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European Mathematical Society Publishing House
2021-10-15 00:06:02
216
https://www.ems-ph.org/meta/bmeta-stream.php?update_since=2021-10-15
Dynamic Optimization for Beginners
With Prerequisites and Applications
Piermarco
Cannarsa
Università di Roma “Tor Vergata”, Italy
Filippo
Gazzola
Politecnico di Milano, Italy
Calculus of variations and optimal control; optimization
Operations research, mathematical programming
49-01, 49L20, 49N90, 90C39
Calculus of variations
Optimization
dynamic optimization, calculus of variations, optimal control, dynamic programming, Pontryagin minimum principle, Hamilton–Jacobi–Bellman equation
Nowadays, optimization problems for dynamical systems enter more and more the basic knowledge required to approach applied sciences, engineering, and social sciences. On the other hand, even the results of most common use in this field, known as dynamic optimization, assume familiarity with an advanced mathematical background which may at times be perceived as discouraging. The main purpose of this book is to give interested readers a friendly introduction to this subject, providing them with the essential notions needed to handle most of the concrete situations they will be faced with. The main topics covered are calculus of variations, optimal control theory, and dynamic programming. As explained above, the book aims to be of interest both to mathematics students and to students or researchers in other disciplines such as economics and data science, who wish to use dynamic optimization in a conscientious way. The contents are self-contained and all the prerequisites may be found in the first chapters. Many applications are discussed and a large number of examples and exercises, both proposed and solved, complements the theory.
10
11
2021
978-3-98547-012-9
978-3-98547-512-4
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/ETB/23
https://www.ems-ph.org/doi/10.4171/ETB/23
EMS Textbooks in Mathematics
An Invitation to Optimal Transport, Wasserstein Distances, and Gradient Flows
Alessio
Figalli
ETH Zürich, Switzerland
Federico
Glaudo
ETH Zürich, Switzerland
Calculus of variations and optimal control; optimization
Measure and integration
Partial differential equations
Probability theory and stochastic processes
49Q22; 60B05, 28A33, 35A15, 35Q35, 49N15, 28A50
Mathematics and science
optimal transport, Wasserstein distance, duality, gradient flows, measure theory, displacement convexity
This book provides a self-contained introduction to optimal transport, and it is intended as a starting point for any researcher who wants to enter into this beautiful subject. The presentation focuses on the essential topics of the theory: Kantorovich duality, existence and uniqueness of optimal transport maps, Wasserstein distances, the JKO scheme, Otto’s calculus, and Wasserstein gradient flows. At the end, a presentation of some selected applications of optimal transport is given. The book is suitable for a course at the graduate level, and also includes an appendix with a series of exercises along with their solutions.
8
2
2021
978-3-98547-010-5
978-3-98547-510-0
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/ETB/22
https://www.ems-ph.org/doi/10.4171/ETB/22
EMS Textbooks in Mathematics
Homogenized Models of Suspension Dynamics
Evgen Ya.
Khruslov
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, Kharkiv
Partial differential equations
Fluid mechanics
35Q35, 76M50, 76T20, 35B27
Mathematics and science
suspension, asymptotic behavior of solutions, mesoscopic characteristics of microstructure, homogenized equations, frozen particles mode, filtering particles mode, cell problem
This book studies the motion of suspensions, that is, of mixtures of a viscous incompressible fluid with small solid particles that can interact with each other through forces of non-hydrodynamic origin. In view of the complexity of the original (microscopic) system of equations that describe such phenomena, which appear both in nature and in engineering processes, the problem is reduced to a macroscopic description of the motion of mixtures as an effective continuous medium. The focus is on developing mathematical methods for constructing such homogenized models for the motion of suspensions with an arbitrary distribution of solid particles in a fluid. In particular, the results presented establish that depending on the concentration of the solid phase of the mixture, the motion of suspensions can occur in two qualitatively different modes: that of frozen or of filtering particles. Being one of the first mathematically rigorous treatises on suspensions from the viewpoint of homogenization theory, this book will be useful to graduate students and researchers in applied analysis and partial differential equations as well as to physicists and engineers interested in the theory of complex fluids with microstructure.
11
22
2021
978-3-98547-009-9
978-3-98547-509-4
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/ETM/34
https://www.ems-ph.org/doi/10.4171/ETM/34
EMS Tracts in Mathematics
34
Orthogonal Polynomials and Linear Functionals
An Algebraic Approach and Applications
Juan Carlos
García-Ardila
Universidad Politécnica de Madrid, Spain
Francisco
Marcellán
Universidad Carlos III de Madrid, Spain
Misael E.
Marriaga
Universidad Rey Juan Carlos, Madrid, Spain
Special functions
Linear and multilinear algebra; matrix theory
Ordinary differential equations
Fourier analysis
33-01, 33C45, 42C05, 15A23, 33C47, 34B24, 65D30
Mathematics and science
linear functionals, orthogonal polynomials, Jacobi matrices, zeros of orthogonal polynomials, Gauss quadrature formulas, spectral transformations, matrix factorization, semiclassical linear functionals, Askey scheme
This book presents an introduction to orthogonal polynomials, with an algebraic flavor, based on linear functionals defining the orthogonality and the Jacobi matrices associated with them. Basic properties of their zeros as well as quadrature rules are discussed. A key point is the analysis of those functionals satisfying Pearson equations (semiclassical case) and the hierarchy based on their class. The book's structure reflects the fact that its content is based on a set of lectures delivered by one of the authors at the first Orthonet Summer School in Seville, Spain in 2016. The presentation of the material is self-contained and will be valuable to students and researchers interested in a novel approach to the study of orthogonal polynomials, focusing on their analytic properties.
8
2
2021
978-3-98547-008-2
978-3-98547-508-7
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/ELM/33
https://www.ems-ph.org/doi/10.4171/ELM/33
EMS Series of Lectures in Mathematics
2523-5176
2523-5184
Partial Differential Equations, Spectral Theory, and Mathematical Physics
The Ari Laptev Anniversary Volume
Pavel
Exner
Czech Technical University, Prague and Czech Academy of Sciences, Řež, Czech Republic
Rupert
Frank
California Institute of Technology, Pasadena, USA and Ludwig-Maximilians-Universität München, Germany
Fritz
Gesztesy
Baylor University, Waco, USA
Helge
Holden
Norwegian University of Science and Technology, Trondheim, Norway
Timo
Weidl
Universität Stuttgart, Germany
Ordinary differential equations
Partial differential equations
Operator theory
34L15, 34L40, 35J10, 35P05, 35P15, 35P20, 44A10, 47A10, 47A63, 47B28, 47F10, 81Q10
Mathematics and science
Friedrichs inequality, Hardy inequality, Lieb–Thirring inequality, Feshbach–Schur map, scattering theory, Wehrl-type entropy inequalities, electron density estimates, stability of matter, Bose–Einstein condensation, heat kernel estimates, Euler–Bardina equations, nonlinear Schrödinger equation, Brezis–Nirenberg problem, d-bar problem, Bogoliubov theory, wave packet evolution
This volume is dedicated to Ari Laptev on the occasion of his 70th birthday. It collects contributions by his numerous colleagues sharing with him research interests in analysis and spectral theory. In brief, the topics covered include Friedrichs, Hardy, and Lieb–Thirring inequalities, eigenvalue bounds and asymptotics, Feshbach–Schur maps and perturbation theory, scattering theory and orthogonal polynomials, stability of matter, electron density estimates, Bose–Einstein condensation, Wehrl-type entropy inequalities, Bogoliubov theory, wave packet evolution, heat kernel estimates, homogenization, d-bar problems, Brezis–Nirenberg problems, the nonlinear Schrödinger equation in magnetic fields, classical discriminants, and the two-dimensional Euler–Bardina equations. In addition, Ari’s multifaceted service to the mathematical community is also touched upon. Altogether the volume presents a collection of research articles which will be of interest to any active scientist working in one of the above mentioned fields.
6
15
2021
978-3-98547-007-5
978-3-98547-507-0
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/ECR/18
https://www.ems-ph.org/doi/10.4171/ECR/18
EMS Series of Congress Reports
2523-515X
2523-5168
18
A non-existence result for a generalized radial Brezis–Nirenberg problem
Rafael
Benguria
Pontificia Universidad Católica de Chile, Santiago de Chile, Chile
Soledad
Benguria
University of Wisconsin, Madison, USA
Brezis–Nirenberg problem, hyperbolic space, non-existence of solutions, Pohozaev identity, Hardy inequality
Partial differential equations
We develop a new method for estimating the region of the spectral parameter of a generalized Brezis–Nirenberg problem for which there are no, non-trivial, smooth solutions. This new method combines the standard Rellich–Pohozaev argument with a “Hardy-type” inequality for bounded domains.
1
15
1
10.4171/ECR/18-1/1
https://www.ems-ph.org/doi/10.4171/ECR/18-1/1
Friedrichs-type inequalities in arbitrary domains
Andrea
Cianchi
Università di Firenze, Italy
Vladimir
Maz'ya
Linköping University, Sweden; University of Liverpool, UK; RUDN University, Moscow, Russia
Sobolev inequalities, irregular domains, boundary traces, symmetric gradient, Lorentz norms, Orlicz norms
Functional analysis
A collection of first- and second-order inequalities for Sobolev functions, involving optimal norms, in arbitrary domains in the Euclidean space is offered. The norms in the relevant Sobolev spaces depend on the highest-order derivatives of functions and on their traces on the boundary of the domain. This allows for constants in the inequalities which are independent of the geometry of the domain. Sobolev spaces of vector-valued functions, defined in terms of their symmetric gradient, are also considered. The results presented rely on a general theory developed in our earlier papers [4] and [5].
17
38
1
10.4171/ECR/18-1/2
https://www.ems-ph.org/doi/10.4171/ECR/18-1/2
Ari Laptev and the Journal of Spectral Theory
E. Brian
Davies
King's College London, UK
39
40
1
10.4171/ECR/18-1/3
https://www.ems-ph.org/doi/10.4171/ECR/18-1/3
Critical magnetic field for 2D magnetic Dirac–Coulomb operators and Hardy inequalities
Jean
Dolbeault
Université de Paris-Dauphine, France
Maria
Esteban
Université Paris-Dauphine, France
Michael
Loss
Georgia Institute of Technology, Atlanta, USA
Aharonov–Bohm magnetic potential, magnetic Dirac operator, Coulomb potential, critical magnetic field, self-adjoint operators, eigenvalues, ground state energy, Hardy inequality, Wirtinger derivatives, Pauli operator
Quantum theory
Functional analysis
This paper is devoted to the study of the two-dimensional Dirac–Coulomb operator in presence of an Aharonov–Bohm external magnetic potential. We characterize the highest intensity of the magnetic field for which a two-dimensional magnetic Hardy inequality holds. Up to this critical magnetic field, the operator admits a distinguished self-adjoint extension and there is a notion of ground state energy, defined as the lowest eigenvalue in the gap of the continuous spectrum.
41
63
1
10.4171/ECR/18-1/4
https://www.ems-ph.org/doi/10.4171/ECR/18-1/4
The Feshbach–Schur map and perturbation theory
Geneviève
Dusson
Université de Bourgogne Franche-Comté, Besançon, France
Israel Michael
Sigal
University of Toronto, Canada
Benjamin
Stamm
RWTH Aachen University, Germany
Perturbation theory, spectrum, Feshbach–Schur map, Schrödinger operator, atomic systems, Helium-type ions, ground state
Operator theory
Partial differential equations
Quantum theory
This paper deals with perturbation theory for discrete spectra of linear operators. To simplify exposition, we consider here self-adjoint operators. This theory is based on the Feshbach–Schur map and it has advantages with respect to the standard perturbation theory in three aspects: (a) it readily produces rigorous estimates on eigenvalues and eigenfunctions with explicit constants; (b) it is compact and elementary (it uses properties of norms and the fundamental theorem of algebra about solutions of polynomial equations); and (c) it is based on a self-contained formulation of a fixed point problem for the eigenvalues and eigenfunctions, allowing for easy iterations. We apply our abstract results to obtain rigorous bounds on the ground states of Helium-type ions.
65
88
1
10.4171/ECR/18-1/5
https://www.ems-ph.org/doi/10.4171/ECR/18-1/5
Adiabatic and non-adiabatic evolution of wave packets and applications to initial value representations
Clotilde
Fermanian Kammerer
Université Paris-Est, France; Université Gustave Eiffel, Marne-la-Vallée, France
Caroline
Lasser
TU München, Garching, Germany
Didier
Robert
Université de Nantes, France
Time-dependent Schrödinger equation, eigenvalue crossing, wave packets, initial value representation
Partial differential equations
Quantum theory
We review some recent results obtained for the time evolution of wave packets for systems of equations of pseudo-differential type, including Schrödinger ones, and discuss their application to the approximation of the associated unitary propagator. We start with scalar equations, propagation of coherent states, and applications to the Herman–Kluk approximation. Then we discuss the extension of these results to systems with eigenvalues of constant multiplicity or with smooth crossings.
89
113
1
10.4171/ECR/18-1/6
https://www.ems-ph.org/doi/10.4171/ECR/18-1/6
Length scales for BEC in the dilute Bose gas
Søren
Fournais
University of Aarhus, Denmark; Institute for Advanced Study Princeton, USA
Many-body quantum mechanics, Dilute Bose gases, Bose–Einstein condensation, Bogolubov theory
Quantum theory
We give a short proof of Bose Einstein Condensation of dilute Bose gases on length scales much longer than the Gross–Pitaevskii scale.
115
133
1
10.4171/ECR/18-1/7
https://www.ems-ph.org/doi/10.4171/ECR/18-1/7
The periodic Lieb–Thirring inequality
Rupert
Frank
Ludwig-Maximilians-Universität München, Germany; California Institute of Technology Pasadena, USA
David
Gontier
Université Paris-Dauphine, France
Mathieu
Lewin
Université Paris-Dauphine, France
Lieb–Thirring inequality, periodic Schrödinger operators
Quantum theory
We discuss the Lieb–Thirring inequality for periodic systems, which has the same optimal constant as the original inequality for finite systems. This allows us to formulate a new conjecture about the value of its best constant. To demonstrate the importance of periodic states, we prove that the 1D Lieb–Thirring inequality at the special exponent $\gamma=\frac{3}{2}$ admits a one-parameter family of periodic optimizers, interpolating between the one-bound state and the uniform potential. Finally, we provide numerical simulations in 2D which support our conjecture that optimizers could be periodic.
135
154
1
10.4171/ECR/18-1/8
https://www.ems-ph.org/doi/10.4171/ECR/18-1/8
Semiclassical asymptotics for a class of singular Schrödinger operators
Rupert
Frank
Ludwig-Maximilians-Universität München, Germany; California Institute of Technology Pasadena, USA
Simon
Larson
California Institute of Technology, Pasadena, USA
Schrödinger operator, semiclassical asymptotics
Partial differential equations
Let $\Omega \subset \mathbb{R}^d$ be bounded with $C^1$ boundary. In this paper we consider Schrödinger operators $-\Delta+ W$ on $\Omega$ with $W(x)\approx\operatorname{dist}(x, \partial\Omega)^{-2}$ as $\operatorname{dist}(x, \partial\Omega)\to 0$. Under weak assumptions on $W$ we derive a two-term asymptotic formula for the sum of the eigenvalues of such operators.
155
176
1
10.4171/ECR/18-1/9
https://www.ems-ph.org/doi/10.4171/ECR/18-1/9
On the spectral properties of the Bloch–Torrey equation in infinite periodically perforated domains
Denis
Grebenkov
École Polytechnique, Palaiseau, France
Bernard
Helffer
Université de Nantes, France
Nicolas
Moutal
École Polytechnique, Palaiseau, France
Bloch–Torrey equation, Floquet theory, non-self-adjoint operators
Operator theory
We investigate spectral and asymptotic properties of the particular Schrödinger operator (also known as the Bloch–Torrey operator), $-\Delta + i g x$, in infinite periodically perforated domains of $\mathbb{R}^d$. We consider Dirichlet realizations of this operator and formalize a numerical approach proposed in [17] for studying such operators. In particular, we discuss the existence of the spectrum of this operator and its asymptotic behavior as $g\to \infty$.
177
196
1
10.4171/ECR/18-1/10
https://www.ems-ph.org/doi/10.4171/ECR/18-1/10
Counting bound states with maximal Fourier multipliers
Dirk
Hundertmark
Karlsruhe Institute of Technology (KIT), Germany; University of Illinois at Urbana-Champaign, USA
Peer Christian
Kunstmann
Karlsruher Institut für Technologie (KIT), Germany
Tobias
Ried
Max-Planck Institut für Mathematik in den Naturwissenschaften, Leipzig, Germany
Semjon
Vugalter
Karlsruhe Institute of Technology (KIT), Germany
CLR inequality, bound states, maximal Fourier multipliers
Partial differential equations
Quantum theory
We report on a version of Cwikel’s proof of the famous Cwikel–Lieb–Rozenblum (CLR) inequality which highlights the connection of the CLR inequality to maximal Fourier multipliers. This new approach enables us to get a constant at least ten times better than Cwikels in all dimensions. In dimensions $d\geq 5$ our results are better than all previously known ones.
197
207
1
10.4171/ECR/18-1/11
https://www.ems-ph.org/doi/10.4171/ECR/18-1/11
Sharp dimension estimates of the attractor of the damped 2D Euler–Bardina equations
Alexei
Ilyin
Keldysh Institute of Applied Mathematics, Moscow, Russian Federation
Sergey
Zelik
Keldysh Institute of Applied Mathematics, Moscow, Russia; Lanzhou University, China; University of Surrey, Guildford, UK
Damped Euler–Bardina equations, $\alpha$ models, attractors, dimension estimates
Partial differential equations
Dynamical systems and ergodic theory
We prove existence of the global attractor of the damped and driven 2D Euler–Bardina equations on the torus and give an explicit two-sided estimate of its dimension that is sharp as $\alpha\to 0^+$.
209
229
1
10.4171/ECR/18-1/12
https://www.ems-ph.org/doi/10.4171/ECR/18-1/12
Upper estimates for the electronic density in heavy atoms and molecules
Victor
Ivrii
University of Toronto, Canada
Electronic density, Thomas–Fermi approximation
Partial differential equations
Quantum theory
We derive an upper estimate for the electronic density $\rho_\Psi(x)$ in heavy atoms and molecules. While not sharp, on the distances $\gtrsim Z^{-1}$ from the nuclei it is still better than the known estimate $CZ^3$ ($Z$ is the total charge of the nuclei, $Z\asymp N$ the total number of electrons).
231
245
1
10.4171/ECR/18-1/13
https://www.ems-ph.org/doi/10.4171/ECR/18-1/13
Non-linear Schrödinger equation in a uniform magnetic field
Thomas
Kieffer
Georgia Institute of Technology, Atlanta, USA
Michael
Loss
Georgia Institute of Technology, Atlanta, USA
Non-linear magnetic Schrödinger equation, blowup, magnetic Strichartz inequalities
Partial differential equations
The aim of this paper is to study, in dimensions 2 and 3, the pure-power nonlinear Schrödinger equation with an external uniform magnetic field included. In particular, we derive a general criteria on the initial data and the power of the non-linearity so that the corresponding solution blows up in finite time, and we show that the time for blow up to occur decreases as the strength of the magnetic field increases. In addition, we also discuss some observations about Strichartz estimates in two dimensions for the Mehler kernel, as well as similar blow-up results for the non-linear Pauli equation.
247
265
1
10.4171/ECR/18-1/14
https://www.ems-ph.org/doi/10.4171/ECR/18-1/14
Large $|k|$ behavior of d-bar problems for domains with a smooth boundary
Christian
Klein
Université de Bourgogne Franche-Comté, Dijon, France
Johannes
Sjöstrand
Université de Bourgogne Franche-Comté, Dijon, France
Nikola
Stoilov
Université de Bourgogne Franche-Comté, Dijon, France
d-bar problems, Dirac systems, semiclassical limit, smooth boundaries
Quantum theory
Functions of a complex variable
In this work we study the large $|k|$ behavior of complex geometric optics solutions to a system of d-bar equations for a potential being the characteristic function of a strictly convex set with smooth boundary, by using almost holomorphic functions. This is an extension of our previous work where we consider sets with real-analytic boundary.
267
275
1
10.4171/ECR/18-1/15
https://www.ems-ph.org/doi/10.4171/ECR/18-1/15
Heat kernel estimates for two-dimensional relativistic Hamiltonians with magnetic field
Hynek
Kovařík
Università degli Studi di Brescia, Italy
Relativistic Hamiltonian, magnetic field, heat kernel
Operator theory
Partial differential equations
We study the long time behavior of the heat kernel generated by relativistic Hamiltonians with radial magnetic field. The main result of the paper shows how the magnetic field accelerates the pointwise decay of the heat kernel. The case of an Aharonov– Bohm magnetic field is discussed as well.
277
288
1
10.4171/ECR/18-1/16
https://www.ems-ph.org/doi/10.4171/ECR/18-1/16
A version of Watson lemma for Laplace integrals and some applications
Stanislas
Kupin
Université de Bordeaux I, Talence, France
Sergey
Naboko
St. Petersburg University, Russian Federation
Asymptotic behavior of a Laplace integral, generalized Watson lemma, special class of functions “slowly decaying to zero” at the origin
Operator theory
Functions of a complex variable
Fourier analysis
Let $f:\mathbb{R}_+\to \mathbb{C}$ be a bounded measurable function. Suppose that $f(t)\to 0$ at logarithmic (or $k$-logarithmic) rate as $t\to 0+$. We consider the Laplace integral of the function $f$, i.e., $$ I_n=\int^\infty_0 f(t)e^{-nt}\,dt $$ and obtain its asymptotics for $n\to+\infty$, which is a version of the classical Watson’s lemma for the integral. Actually, the result is proved for a larger class of “slowly oscillating” functions satisfying some mild regularity conditions.
289
300
1
10.4171/ECR/18-1/17
https://www.ems-ph.org/doi/10.4171/ECR/18-1/17
Wehrl-type coherent state entropy inequalities for SU(1,1) and its $AX+B$ subgroup
Elliott
Lieb
Princeton University, USA
Jan Philip
Solovej
University of Copenhagen, Denmark
Coherent states, affine group, $AX+B$ group
Quantum theory
Topological groups, Lie groups
We discuss the Wehrl-type entropy inequality conjecture for the group $\mathrm{SU}(1,1)$ and for its subgroup $AX+B$ (or affine group), their representations on $L^2(\mathbb{R}_+)$, and their coherent states. For $AX+B$ the Wehrl-type conjecture for $L^p$-norms of these coherent states (also known as the Renyi entropies) is proved in the case that $p$ is an even integer. We also show how the general $AX+B$ case reduces to an unsolved problem about analytic functions on the upper half-plane and the unit disk.
301
314
1
10.4171/ECR/18-1/18
https://www.ems-ph.org/doi/10.4171/ECR/18-1/18
Blow-ups for the Horn–Kapranov parametrization of the classical discriminant
Evgeny
Mikhalkin
Siberian Federal University, Krasnoyarsk, Russian Federation
Vitaly
Stepanenko
Siberian Federal University, Krasnoyarsk, Russian Federation
Avgust
Tsikh
Siberian Federal University, Krasnoyarsk, Russian Federation
Discriminant, Horn–Kapranov parametrization, Newton polytope, logarithmic Gauss map, blow-up
Several complex variables and analytic spaces
The Horn–Kapranov parametrizations describe the singular sets of hypergeometric functions in several variables. These parametrizations are inverses of logarithmic Gauss maps for A-discriminants. In this paper we demonstrate that, despite the multivalued nature of the indicated parametrizations, their blow-ups properties are the same as for single-valued meromorphic mappings. As an application, a new proof of factorization identities for the classical discriminant is given.
315
329
1
10.4171/ECR/18-1/19
https://www.ems-ph.org/doi/10.4171/ECR/18-1/19
Eigenvalue estimates and asymptotics for weighted pseudodifferential operators with singular measures in the critical case
Grigori
Rozenblum
Chalmers University of Technology, Göteborg, Sweden; St. Petersburg State University; Russian Federation
Eugene
Shargorodsky
King's College London, UK; Technische Universität Dresden, Germany
Eigenvalue estimates, eigenvalue asymptotics, pseudodifferential operators, singular measures
Operator theory
Global analysis, analysis on manifolds
We consider self-adjoint operators of the form $\mathbf{T}_{P,\mathfrak{A}}=\mathfrak{A}^* P \mathfrak{A}$ in a domain $\Omega\subset\mathbb{R}^\mathbf{N}$, where $\mathfrak{A}$ is an order $-l=-\frac{\mathbf{N}}{2}$ pseudodifferential operator in $\Omega$ and $P$ is a signed Borel measure with compact support in $\Omega$. Measure $P$ may contain singular component. For a wide class of measures we establish eigenvalue estimates for operator $\mathbf{T}_{P,\mathfrak{A}}$. In case of measure $P$ being absolutely continuous with respect to the Hausdorff measure on a Lipschitz surface of an arbitrary dimension, we find the eigenvalue asymptotics. The order of eigenvalue estimates and asymptotics does not depend on dimensional characteristics of the measure, in particular, on the dimension of the surface supporting the measure.
331
354
1
10.4171/ECR/18-1/20
https://www.ems-ph.org/doi/10.4171/ECR/18-1/20
Relations between two parts of the spectrum of a Schrödinger operator and other remarks on the absolute continuity of the spectrum in a typical case
Oleg
Safronov
University of North Carolina at Charlotte, USA
Schrödinger operators, absolutely continuous spectrum, discrete spectrum
Quantum theory
Operator theory
We will discuss relations between different parts of spectra of differential operators. In particular, we will see that negative and positive spectra of Schrödinger operators are related to each other. However, there is a stipulation: one needs to consider two operators one of which is opened from the other by flipping the sign of the potential at each point x. If one knows only that the negative spectra of the two operators are discrete, then their positive spectra do not have gaps. If one knows more about the rate of accumulation of the discrete negative eigenvalues to zero, then one can say more about the absolutely continuous component of the positive spectrum. The second part of this article contains a discussion of spectral properties of a family of Schrödiger operators depending on a real parameter $t$. The results claim that the absolutely continuous spectrum of an operator of this family is essentially supported by the positive half-line for almost every $t$.
355
365
1
10.4171/ECR/18-1/21
https://www.ems-ph.org/doi/10.4171/ECR/18-1/21
Bogoliubov theory for many-body quantum systems
Benjamin
Schlein
Universität Zürich, Switzerland
Many-body quantum mechanics, Bogoliubov theory, Bose–Einstein condensates, mean-field limit, Hartree–Fock theory, Fröhlich polaron
Partial differential equations
Operator theory
Quantum theory
Statistical mechanics, structure of matter
We review some recent applications of rigorous Bogoliubov theory. We show how Bogoliubov theory can be used to approximate quantum fluctuations, both in the analysis of the energy spectrum and in the study of the dynamics of many-body quantum systems.
367
388
1
10.4171/ECR/18-1/22
https://www.ems-ph.org/doi/10.4171/ECR/18-1/22
A statistical theory of heavy atoms: Asymptotic behavior of the energy and stability of matter
Heinz
Siedentop
Ludwig-Maximilians-Universität München, Germany; Munich Center for Quantum Science and Technology (MCQST), Germany
Heavy atom, asymptotic behavior of the ground energy, Engel–Dreizler functional
Quantum theory
We give the asymptotic behavior of the ground state energy of Engel’s and Dreizler’s relativistic Thomas–Fermi–Weizsäcker–Dirac functional for heavy atoms for fixed ratio of the atomic number and the velocity of light. Using a variation of the lower bound, we show stability of matter.
389
403
1
10.4171/ECR/18-1/23
https://www.ems-ph.org/doi/10.4171/ECR/18-1/23
Homogenization of the higher-order Schrödinger-type equations with periodic coefficients
Tatiana
Suslina
St. Petersburg State University, Russian Federation
Homogenization, periodic differential operators, operator error estimates
Partial differential equations
In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider a matrix strongly elliptic differential operator $A_\varepsilon$ of order $2p$, $p \geqslant 2$. The operator $A_\varepsilon$ is given by $A_\varepsilon = b(\mathbf{D})^* g(\frac{\mathbf{x}}{\varepsilon}) b(\mathbf{D})$, $\varepsilon >0$, where $g(\mathbf{x})$ is a periodic, bounded, and positive definite matrix-valued function, and $b(\mathbf{D})$ is a homogeneous differential operator of order $p$. We prove that, for fixed $\tau \in \mathbb{R}$ and $\varepsilon \to 0$, the operator exponential $e^{-i \tau A_\varepsilon}$ converges to $e^{-i \tau A^0}$ in the norm of operators acting from the Sobolev space $H^s(\mathbb{R}^d;\mathbb{C}^n)$ (with a suitable $s$) into $L_2(\mathbb{R}^d;\mathbb{C}^n)$. Here $A^0$ is the effective operator. Sharp-order error estimate is obtained. The results are applied to homogenization of the Cauchy problem for the Schrödinger-type equation $i \partial_\tau \mathbf{u}_\varepsilon = A_\varepsilon \mathbf{u}_\varepsilon + \mathbf{F}$, $\mathbf{u}_\varepsilon\vert_{\tau=0} = \boldsymbol{\phi}$.
405
426
1
10.4171/ECR/18-1/24
https://www.ems-ph.org/doi/10.4171/ECR/18-1/24
Trace formulas for the modified Mathieu equation
Leon
Takhtajan
SUNY at Stony Brook, USA; Euler International Mathematical Institute, St. Petersburg, Russian Federation
Fredholm determinant, Green–Liouville method, Mathieu equation, radial and one-dimensional Schrödinger operator, Riccati equation, trace identities
Operator theory
Ordinary differential equations
For the radial and one-dimensional Schrödinger operator $H$ with growing potential $q(x)$ we outline a method of obtaining the trace identities – an asymptotic expansion of the Fredholm determinant $\mathrm{det}_{F}(H-\lambda I)$ as $\lambda\to-\infty$. As an illustrating example, we consider Schrödinger operator with the potential $q(x)=2\cosh 2x$, associated with the modified Mathieu equation.
427
443
1
10.4171/ECR/18-1/25
https://www.ems-ph.org/doi/10.4171/ECR/18-1/25
Eigenvalue accumulation and bounds for non-selfadjoint matrix differential operators related to NLS
Christiane
Tretter
Universität Bern, Switzerland
Eigenvalue bounds, eigenvalue accumulation, non-linear Schrödinger operator, NLS, operator matrix, non-selfadjoint operator, Krein space
Partial differential equations
Operator theory
We establish results on the accumulation and location of the non-real spectrum of non-selfadjoint matrix differential operators arising in the study of non-linear Schrödinger equations (NLS) in $\mathbb{R}^d$. In particular, without restrictions on the decay rate of the potentials to $0$ at $\infty$, we show that the non-real spectrum cannot accumulate anywhere on the real axis. Under some weak assumptions satisfied, e.g., by $L_p$-potentials with $p>\frac d2$, $p\ge 2$, we prove that there are only finitely many non-real eigenvalues and that the non-real eigenvalues are located in a bounded lens-shaped region centered at the origin. Our key tool to prove this is a recent result on the existence of $\mathcal{J}$-semi-definite invariant subspaces for $\mathcal{J}$-selfadjoint operators in Krein spaces as well as abstract operator matrix methods.
445
456
1
10.4171/ECR/18-1/26
https://www.ems-ph.org/doi/10.4171/ECR/18-1/26
Scattering theory for Laguerre operators
Dmitri
Yafaev
Université de Rennes I, France; St. Petersburg State University, Russian Federation
Jacobi operators, Laguerre polynomials, asymptotic formulas
Special functions
Difference and functional equations
Operator theory
We study Jacobi operators $J_{p}$, $p> -1$, whose eigenfunctions are Laguerre polynomials. All operators $J_{p}$ have absolutely continuous simple spectra coinciding with the positive half-axis. This fact, however, by no means imply that the wave operators for the pairs $J_{p}$, $J_{q}$ where $p\neq q$ exist. Our goal is to show that, nevertheless, this is true and to find explicit expressions for these wave operators. We also study the time evolution of $(e^{-J t} f)_{n}$ as $|t|\to\infty$ for Jacobi operators $J$ whose eigenfunctions are different classical polynomials. For Laguerre polynomials, it turns out that the evolution $(e^{-J_{p} t} f)_{n}$ is concentrated in the region where $n\sim t^2$ instead of $n\sim |t|$ as happens in standard situations. As a by-product of our considerations, we obtain universal relations between amplitudes and phases in asymptotic formulas for general orthogonal polynomials.
457
478
1
10.4171/ECR/18-1/27
https://www.ems-ph.org/doi/10.4171/ECR/18-1/27
Handbook of Automata Theory
Volume I. Theoretical Foundations Volume II. Automata in Mathematics and Selected Applications
Jean-Éric
Pin
Université de Paris and CNRS, France
Computer science
68Q45
Mathematics and science
finite automata, Hopcroft's algorithm, regular languages, regular expressions, finite transducers, weighted automata, automata on infinite words, automata on trees, picture languages, algorithmic learning, descriptional complexity, Boolean circuits, synchronizing automata, road colouring problem, varieties of languages, profinite topology, descriptive set theory, equational theories, language equations, forest algebras, free groups, Stallings automata, automatic groups, self-similar groups, automatic sequences, numeration systems, Cobham's theorem, symbolic dynamics, synchronous automata, finite model theory, fractal image generation, communicating automata, model-checking, Church's problem, distributed synthesis, timed automata, recursion schemes, Markov decision processes, infinite-state systems, natural language processing, temporal logic, branching time logic, quantum finite automata
Automata theory is a subject of study at the crossroads of mathematics, theoretical computer science, and applications. In its core it deals with abstract models of systems whose behaviour is based on transitions between states, and it develops methods for the description, classification, analysis, and design of such systems. The Handbook of Automata Theory gives a comprehensive overview of current research in automata theory, and is aimed at a broad readership of researchers and graduate students in mathematics and computer science. Volume I is divided into three parts. The first part presents various types of automata: automata on words, on infinite words, on finite and infinite trees, weighted and maxplus automata, transducers, and two-dimensional models. Complexity aspects are discussed in the second part. Algebraic and topological aspects of automata theory are covered in the third part. Volume II consists of two parts. The first part is dedicated to applications of automata in mathematics: group theory, number theory, symbolic dynamics, logic, and real functions. The second part presents a series of further applications of automata theory such as message-passing systems, symbolic methods, synthesis, timed automata, verification of higher-order programs, analysis of probabilistic processes, natural language processing, formal verification of programs and quantum computing. The two volumes comprise a total of thirty-nine chapters, with extensive references and individual tables of contents for each one, as well as a detailed subject index.
9
13
2021
978-3-98547-006-8
978-3-98547-506-3
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/Automata
https://www.ems-ph.org/doi/10.4171/Automata
Finite automata
Jean-Éric
Pin
Université de Paris et CNRS, France
3
38
1
10.4171/Automata-1/1
https://www.ems-ph.org/doi/10.4171/Automata-1/1
Automata and rational expressions
Jacques
Sakarovitch
Université de Paris et CNRS, France
39
78
1
10.4171/Automata-1/2
https://www.ems-ph.org/doi/10.4171/Automata-1/2
Finite transducers and rational transductions
Tero
Harju
University of Turku, Finland
Juhani
Karhumäki
University of Turku, Finland
79
111
1
10.4171/Automata-1/3
https://www.ems-ph.org/doi/10.4171/Automata-1/3
Weighted automata
Manfred
Droste
Universität Leipzig, Germany
Dietrich
Kuske
Technische Universität Ilmenau, Germany
113
150
1
10.4171/Automata-1/4
https://www.ems-ph.org/doi/10.4171/Automata-1/4
Max-plus automata
Sylvain
Lombardy
Université de Bordeaux et CNRS, Talence, France
Jean
Mairesse
Univ Paris Diderot, Sorbonne Paris Cit, France
151
188
1
10.4171/Automata-1/5
https://www.ems-ph.org/doi/10.4171/Automata-1/5
ω-Automata
Thomas
Wilke
Christian-Albrechts-Universität zu Kiel, Germany
Sven
Schewe
University of Liverpool, UK
189
234
1
10.4171/Automata-1/6
https://www.ems-ph.org/doi/10.4171/Automata-1/6
Automata on finite trees
Christof
Löding
RWTH Aachen, Germany
Wolfgang
Thomas
RWTH Aachen, Germany
235
264
1
10.4171/Automata-1/7
https://www.ems-ph.org/doi/10.4171/Automata-1/7
Automata on infinite trees
Christof
Löding
RWTH Aachen, Germany
265
302
1
10.4171/Automata-1/8
https://www.ems-ph.org/doi/10.4171/Automata-1/8
Two-dimensional models
Stefano
Crespi Reghizzi
Politecnico di Milano, Italy
Dora
Giammarresi
Università di Roma, Italy
Violetta
Lonati
Università degli Studi di Milano, Italy
303
333
1
10.4171/Automata-1/9
https://www.ems-ph.org/doi/10.4171/Automata-1/9
Minimisation of automata
Jean
Berstel
Université Paris-Est Marne-la-Vallée, France
Luc
Boasson
Université de Paris et CNRS, France
Olivier
Carton
Université de Paris et CNRS, France
Isabelle
Fagnot
Université Paris-Est Marne-la-Vallée, France
337
373
1
10.4171/Automata-1/10
https://www.ems-ph.org/doi/10.4171/Automata-1/10
Learning algorithms
Henrik
Björklund
Umeå University, Sweden
Johanna
Björklund
Umeå University, Sweden
Wim
Martens
Universität Bayreuth, Germany
375
409
1
10.4171/Automata-1/11
https://www.ems-ph.org/doi/10.4171/Automata-1/11
Descriptional complexity of regular languages
Hermann
Gruber
Knowledgepark GmbH, München, Germany
Markus
Holzer
Universität Giessen, Germany
Martin
Kutrib
Universität Giessen, Germany
411
457
1
10.4171/Automata-1/12
https://www.ems-ph.org/doi/10.4171/Automata-1/12
Enumerating regular expressions and their languages
Hermann
Gruber
Knowledgepark GmbH, München, Germany
Jonathan
Lee
Stanford University, USA
Jeffrey
Shallit
University of Waterloo, Canada
459
491
1
10.4171/Automata-1/13
https://www.ems-ph.org/doi/10.4171/Automata-1/13
Circuit complexity of regular languages
Michal
Koucký
Charles University, Praha, Czechia
493
523
1
10.4171/Automata-1/14
https://www.ems-ph.org/doi/10.4171/Automata-1/14
Černý’s conjecture and the road colouring problem
Jarkko
Kari
University of Turku, Finland
Mikhail
Volkov
Ural Federal University, Ekaterinburg, Russia
525
565
1
10.4171/Automata-1/15
https://www.ems-ph.org/doi/10.4171/Automata-1/15
Varieties
Howard
Straubing
Boston College, Chestnut Hill, USA
Pascal
Weil
Université de Bordeaux et CNRS, Talence, France
569
614
1
10.4171/Automata-1/16
https://www.ems-ph.org/doi/10.4171/Automata-1/16
Profinite topologies
Jorge
Almeida
Faculdade de Ciências, Universidade do Porto, Portugal
Alfredo
Costa
University of Coimbra, Portugal
615
652
1
10.4171/Automata-1/17
https://www.ems-ph.org/doi/10.4171/Automata-1/17
The factorisation forest theorem
Thomas
Colcombet
Université de Paris et CNRS, France
653
693
1
10.4171/Automata-1/18
https://www.ems-ph.org/doi/10.4171/Automata-1/18
Wadge–Wagner hierarchies
Jacques
Duparc
University of Lausanne, Switzerland
695
728
1
10.4171/Automata-1/19
https://www.ems-ph.org/doi/10.4171/Automata-1/19
Equational theories for automata
Zoltán
Ésik
University of Szeged, Hungary
729
764
1
10.4171/Automata-1/20
https://www.ems-ph.org/doi/10.4171/Automata-1/20
Language equations
Michal
Kunc
Masaryk University, Brno, Czechia
Alexander
Okhotin
St. Petersburg State University, Russia
765
799
1
10.4171/Automata-1/21
https://www.ems-ph.org/doi/10.4171/Automata-1/21
Algebra for trees
Mikołaj
Bojańczyk
University of Warsaw, Poland
801
838
1
10.4171/Automata-1/22
https://www.ems-ph.org/doi/10.4171/Automata-1/22
Rational subsets of groups
Laurent
Bartholdi
Georg-August-Universität Göttingen, Germany
Pedro
Silva
Faculdade de Ciências, Universidade do Porto, Portugal
841
869
1
10.4171/Automata-1/23
https://www.ems-ph.org/doi/10.4171/Automata-1/23
Groups defined by automata
Laurent
Bartholdi
Georg-August-Universität Göttingen, Germany
Pedro
Silva
Faculdade de Ciências, Universidade do Porto, Portugal
871
911
1
10.4171/Automata-1/24
https://www.ems-ph.org/doi/10.4171/Automata-1/24
Automata in number theory
Boris
Adamczewski
CNRS, Université de Lyon, Villeurbanne, France
Jason
Bell
University of Waterloo, Canada
913
945
1
10.4171/Automata-1/25
https://www.ems-ph.org/doi/10.4171/Automata-1/25
On Cobham’s theorem
Fabien
Durand
Université de Picardie Jules Verne, Amiens, France
Michel
Rigo
Université de Liège, Belgium
947
986
1
10.4171/Automata-1/26
https://www.ems-ph.org/doi/10.4171/Automata-1/26
Symbolic dynamics
Marie-Pierre
Béal
Université Paris-Est Marne-la-Vallée, France
Jean
Berstel
Université Paris-Est Marne-la-Vallée, France
Søren
Eilers
Københavns Universitet, Denmark
Dominique
Perrin
Université Paris-Est Marne-la-Vallée, France
987
1030
1
10.4171/Automata-1/27
https://www.ems-ph.org/doi/10.4171/Automata-1/27
Automatic structures
Sasha
Rubin
The University of Sydney, Camperdown, Australia
1031
1070
1
10.4171/Automata-1/28
https://www.ems-ph.org/doi/10.4171/Automata-1/28
Automata and finite model theory
Wouter
Gelade
University of Namur, Belgium
Thomas
Schwentick
TU Dortmund University, Germany
1071
1103
1
10.4171/Automata-1/29
https://www.ems-ph.org/doi/10.4171/Automata-1/29
Finite automata, image manipulation, and automatic real functions
Juhani
Karhumäki
University of Turku, Finland
Jarkko
Kari
University of Turku, Finland
1105
1143
1
10.4171/Automata-1/30
https://www.ems-ph.org/doi/10.4171/Automata-1/30
Communicating automata
Dietrich
Kuske
Technische Universität Ilmenau, Germany
Anca
Muscholl
Université de Bordeaux et CNRS, Talence, France
1147
1188
1
10.4171/Automata-1/31
https://www.ems-ph.org/doi/10.4171/Automata-1/31
Symbolic methods and automata
Bernard
Boigelot
Université de Liège, Belgium
1189
1215
1
10.4171/Automata-1/32
https://www.ems-ph.org/doi/10.4171/Automata-1/32
Synthesis with finite automata
Igor
Walukiewicz
Université de Bordeaux et CNRS, Talence, France
1217
1260
1
10.4171/Automata-1/33
https://www.ems-ph.org/doi/10.4171/Automata-1/33
Timed automata
Patricia
Bouyer
Université Paris-Saclay, CNRS, Gif-sur-Yvette, France
1261
1294
1
10.4171/Automata-1/34
https://www.ems-ph.org/doi/10.4171/Automata-1/34
Higher-order recursion schemes and their automata models
Arnaud
Carayol
Université Gustave Eiffel, Marne-la-Vallée Cedex 2, France
Olivier
Serre
Université de Paris et CNRS, France
1295
1341
1
10.4171/Automata-1/35
https://www.ems-ph.org/doi/10.4171/Automata-1/35
Analysis of probabilistic processes and automata theory
Kousha
Etessami
University of Edinburgh, UK
1343
1382
1
10.4171/Automata-1/36
https://www.ems-ph.org/doi/10.4171/Automata-1/36
Natural language parsing
Mark-Jan
Nederhof
University of St Andrews, Scotland
Giorgio
Satta
University of Padua, Padova, Italy
1383
1413
1
10.4171/Automata-1/37
https://www.ems-ph.org/doi/10.4171/Automata-1/37
Verification
Javier
Esparza
Technische Universität München, Germany
Orna
Kupferman
Hebrew University, Jerusalem, Israel
Moshe
Vardi
Rice University, Houston, USA
1415
1456
1
10.4171/Automata-1/38
https://www.ems-ph.org/doi/10.4171/Automata-1/38
Automata and quantum computing
Andris
Ambainis
University of Latvia, Riga, Latvia
Abuzer
Yakaryılmaz
University of Latvia, Riga, Latvia
1457
1493
1
10.4171/Automata-1/39
https://www.ems-ph.org/doi/10.4171/Automata-1/39
Probabilistic Structures in Evolution
Ellen
Baake
Universität Bielefeld, Germany
Anton
Wakolbinger
Goethe-Universität, Frankfurt a.M., Germany
Biology and other natural sciences
Probability theory and stochastic processes
92D15, 60-XX
Mathematics and science
stochastic processes, population genetics, population dynamics, coalescent theory, random trees
The present volume collects twenty-one survey articles about probabilistic aspects of biological evolution. They cover a large variety of topics from the research done within the German Priority Programme SPP 1590. Evolution is a complex phenomenon driven by various processes, such as mutation and recombination of genetic material, reproduction of individuals, and selection of favourable types. These processes all have intrinsically random elements, which give rise to a wealth of phenomena that cannot be explained by deterministic models. Examples of such effects are the loss of genetic variability due to random reproduction and the emergence of random genealogies. The collection is centred around the stochastic processes in population genetics and population dynamics. On the one hand, these are individual-based models of predator-prey and of coevolution type, of adaptive dynamics, or of experimental evolution, considered in the usual forward direction of time. They lead to processes describing the evolution of type frequencies, which may then be analysed via suitable limit theorems. On the other hand, one traces the ancestral lines of individuals back into the past; this leads to random genealogies. Beyond the classical concept of Kingman's coalescent, emphasis is on genealogies with multiple mergers and on ancestral structures that take into account selection, recombination, or migration. The contributions in this volume will be valuable to researchers interested in stochastic processes and their biological applications, or in mathematical population biology.
5
31
2021
978-3-98547-005-1
978-3-98547-505-6
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/ECR/17
https://www.ems-ph.org/doi/10.4171/ECR/17
EMS Series of Congress Reports
2523-515X
2523-5168
Accessibility percolation in random fitness landscapes
Joachim
Krug
Universität zu Köln, Germany
The fitness landscape encodes the mapping of genotypes to fitness and provides a succinct representation of possible trajectories followed by an evolving population. Evolutionary accessibility is quantified by the existence of fitness-monotonic paths connecting far away genotypes. Studies of accessibility percolation use probabilistic fitness landscape models to explore the emergence of such paths as a function of the initial fitness, the parameters of the landscape or the structure of the genotype graph. This chapter reviews these studies and discusses their implications for the predictability of evolutionary processes.
1
22
1
10.4171/ECR/17-1/1
https://www.ems-ph.org/doi/10.4171/ECR/17-1/1
Branching random walks in random environment
Wolfgang
König
Weierstraß-Institut Berlin für Angewandte Analysis und Stochastik, Germany; TU Berlin, Germany
We consider branching particle processes on discrete structures like the hypercube in a random fitness landscape (i.e. random branching/killing rates). The main question is about the location where the main part of the population sits at a late time, if the state space is large. For answering this, we take the expectation with respect to the migration (mutation) and the branching/killing (selection) mechanisms, for fixed rates. This is intimately connected with the parabolic Anderson model, the heat equation with random potential, a model that is of interest in mathematical physics because of the observed prominent effect of intermittency (local concentration of the mass of the solution in small islands). We present several advances in the investigation of this effect, also related to questions inspired from biology.
23
41
1
10.4171/ECR/17-1/2
https://www.ems-ph.org/doi/10.4171/ECR/17-1/2
Microbial populations under selection
Ellen
Baake
Universität Bielefeld, Germany
Anton
Wakolbinger
J. W. Goethe-Universität, Frankfurt a.M., Germany
This chapter gives a synopsis of recent approaches to model and analyse the evolution of microbial populations under selection. The first part reviews two population genetic models of Lenski’s long-term evolution experiment with Escherichia coli, where models aim at explaining the observed curve of the evolution of the mean fitness. The second part describes a model of a host-pathogen system where the population of pathogens experiences balancing selection, migration, and mutation, as motivated by observations of the genetic diversity of HCMV (the human cytomegalovirus) across hosts.
43
68
1
10.4171/ECR/17-1/3
https://www.ems-ph.org/doi/10.4171/ECR/17-1/3
The population genetics of the CRISPR-Cas system in bacteria
Rolf
Backofen
Albert-Ludwigs-Universität Freiburg, Germany
Peter
Pfaffelhuber
Albert-Ludwigs-Universität Freiburg, Germany
The Clustered Regularly Interspaced Short Palindromic Repeats-system (or CRISPR- Cas system) is known as the immune system of bacteria against phages. Albeit the general function is similar across the different CRISPR systems, the known systems are highly diverged, and a classification system is required that identifies the different components and their function. In addition, for any given CRISPR system, a population sample may have different spacers, which calls for a population genetic model for the spacer sequences. In future work, such a model can be used to determine rates for spacer gain and loss depending on the type of the CRISPR system.
69
84
1
10.4171/ECR/17-1/4
https://www.ems-ph.org/doi/10.4171/ECR/17-1/4
Evolution of altruistic defence traits in structured populations
Martin
Hutzenthaler
Universität Duisburg-Essen, Germany
Dirk
Metzler
Ludwig-Maximilans-Universität München, Martinsried, Germany
Defence traits against predators, as for example alarm calls, can be costly for the acting individual and beneficial for others in the same subpopulation. There is a growing literature trying to explain persistence of such altruistic defence traits. In this review we summarise recent progress on a specific individual-based Lotka–Volterra-type predator-prey model with two types (altruists and cheaters) of prey. This dynamic can also be considered as a model for host-parasite interactions, where parasites correspond to predators and hosts correspond to prey. For our analysis of persistence of altruists, we focus on the special case of uniform migration on finitely many demes. We perform two approximation steps: First we let the number of individuals tend to infinity and then we let the number of demes tend to infinity. The central observation for this McKean–Vlasov-type diffusion limit is then that the altruistic defence trait persists in the population if the cost of defence is smaller than a particular model parameter, which can be interpreted as benefit of defence.
85
106
1
10.4171/ECR/17-1/5
https://www.ems-ph.org/doi/10.4171/ECR/17-1/5
Stochastic processes and host-parasite coevolution: Linking coevolutionary dynamics and DNA polymorphism data
Wolfgang
Stephan
Museum für Naturkunde, Berlin, Germany
Aurélien
Tellier
TU München, Freising, Germany
Between-species coevolution, and in particular antagonistic host-parasite coevolution, is a major process shaping within-species diversity. In this paper we investigate the role of various stochastic processes affecting the outcome of the deterministic coevolutionary models. Specifically, we assess (1) the impact of genetic drift and mutation on the maintenance of polymorphism at the interacting loci, and (2) the change in neutral allele frequencies across the genome of both coevolving species due to co-demographic population size changes. We find that genetic drift decreases the likelihood to observe classic balancing selection signatures, and that for most realistic values of the coevolutionary parameters, balancing selection signatures cannot be seen at the host loci. Further, we reveal that contrary to classic expectations, fast changes in parasite population size due to eco-evo feedbacks can be tracked by the allelic site-frequency spectrum measured at several time points. Changes in host population size are, however, less pronounced and thus not observable. Finally, we also review several understudied stochastic processes occurring in host-parasite coevolution which are of importance to predict maintenance of polymorphism at the underlying loci and the genome-wide nucleotide diversity of host and parasite populations.
107
126
1
10.4171/ECR/17-1/6
https://www.ems-ph.org/doi/10.4171/ECR/17-1/6
Stochastic models for adaptive dynamics: Scaling limits and diversity
Anton
Bovier
Rheinische-Friedrich-Wilhelms-Universität Bonn, Germany
I discuss the so-called stochastic individual based model of adaptive dynamics and in particular how different scaling limits can be obtained by taking limits of large populations, small mutation rate, and small effect of single mutations together with appropriate time rescaling. In particular, one derives the trait substitution sequence, polymorphic evolution sequence, and the canonical equation of adaptive dynamics. In addition, I show how the escape from an evolutionary stable condition can occur as a metastable transition.
127
150
1
10.4171/ECR/17-1/7
https://www.ems-ph.org/doi/10.4171/ECR/17-1/7
Genealogies and inference for populations with highly skewed offspring distributions
Matthias
Birkner
Johannes-Gutenberg-Universität Mainz, Germany
Jochen
Blath
TU Berlin, Germany
We review recent progress in the understanding of the role of multiple- and simultan- eous multiple merger coalescents as models for the genealogy in idealised and real populations with exceptional reproductive behaviour. In particular, we discuss models with “skewed offspring distribution” (or under other non-classical evolutionary forces) which lead to multiple merger coalescents in the single locus haploid case, and to simultaneous multiple merger coalescents in the multi-locus diploid case. Further, we discuss inference methods under the infinitely-many sites model which allow both model selection and estimation of model parameters under these coalescents.
151
178
1
10.4171/ECR/17-1/8
https://www.ems-ph.org/doi/10.4171/ECR/17-1/8
Multiple-merger genealogies: Models, consequences, inference
Fabian
Freund
Universität Hohenheim, Stuttgart, Germany
Trees corresponding to Λ- and Ξ-n-coalescents can be both quite similar and funda- mentally different compared to bifurcating tree models based on Kingman’s n-coales- cent. This has consequences for inference of a well-fitting gene genealogy as well as for assessing biological properties of species having such sample genealogies. Here, mathematical properties concerning clade sizes in the tree as well as changes of the tree when the samples are enlarged are highlighted. To be used as realistic genealogy models for real populations, an extension for changing population sizes is discussed.
179
202
1
10.4171/ECR/17-1/9
https://www.ems-ph.org/doi/10.4171/ECR/17-1/9
Diploid populations and their genealogies
Anja
Sturm
Georg-August-Universität Göttingen, Germany
Diploid organisms carry pairs of homologous chromosomes that are inherited from two parents who each contribute exactly one homologous chromosome, while haploid organisms inherit just one copy from a single parent. In this contribution, we summarise classical results on the genealogies of haploid populations with large but fixed total population size, which have also been used as approximations to the diploid case by ignoring the pairing into individuals. We then present recent results on extending the characterisation of the genealogies to analogous, explicitly diploid models. We discuss the implications and illustrate them by means of a number of concrete examples. Lastly, we survey related research works and further questions regarding the modelling of diploid reproduction.
203
222
1
10.4171/ECR/17-1/10
https://www.ems-ph.org/doi/10.4171/ECR/17-1/10
Probabilistic aspects of Λ-coalescents in equilibrium and in evolution
Götz
Kersting
Goethe-Universität Frankfurt am Main, Germany
Anton
Wakolbinger
J. W. Goethe-Universität, Frankfurt a.M., Germany
We present approximation methods that lead to law of large numbers and fluctuation results for functionals of Λ-coalescents, both in the dust-free case and in the case with a dust component. Our focus is on the tree length (or total branch length) and the total external branch length, as well as the time to the most recent common ancestor and the size of the last merger. In the second part we discuss evolving coalescents. For certain Beta-coalescents we analyse fluctuations of a class of functionals on appropriate time scales. Finally, we review results of Gufler on the representation of evolving Λ-coalescents in terms of the lookdown space.
223
246
1
10.4171/ECR/17-1/11
https://www.ems-ph.org/doi/10.4171/ECR/17-1/11
Population genetic models of dormancy
Jochen
Blath
TU Berlin, Germany
Noemi
Kurt
TU Berlin, Germany
In the present article, we investigate the effects of dormancy on an abstract popula- tion genetic level. We first provide a short review of seed bank models in population genetics, and the role of dormancy for the interplay of evolutionary forces in gen- eral, before we discuss two recent paradigmatic models, referring to spontaneous resp. simultaneous switching of individuals between the active and the dormant state. We show that both mechanisms give rise to non-trivial mathematical objects, namely the (continuous) seed bank diffusion and the seed bank diffusion with jumps, as well as their dual processes, the seed bank coalescent and the seed bank coalescent with simultaneous switching.
247
266
1
10.4171/ECR/17-1/12
https://www.ems-ph.org/doi/10.4171/ECR/17-1/12
From high to low volatility: Spatial Cannings with block resampling and spatial Fleming–Viot with seed-bank
Andreas
Greven
Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
Frank
den Hollander
Universiteit Leiden, Netherlands
In neutrally evolving populations subject to resampling and migration on geographic spaces, the longtime behaviour exhibits a dichotomy between clustering of types versus coexistence of types (that is between convergence to monotype equilibria versus multitype equilibria). In the classical setting, which of the two scenarios prevails depends on whether the difference between the positions of two lineages in the dual is a recurrent random walk or a transient random walk. In the present contribution, for two classes of models we present results on how this dichotomy is affected when the resampling has either high volatility or low volatility: (I) The spatial Cannings model with block resampling, where a positive fraction of the individuals in the next generation may inherit the type of a single individual in the previous generation. (II) The spatial Fleming–Viot model with seed-bank, where individuals may become dormant for a while, suspending resampling and migration, until they become active again. We will see that the dichotomy is shifted towards more clustering in class (I) and towards more coexistence in class (II). In particular, in class (II) we will see that for critically recurrent random walks (which are typical for two-dimensional geographic spaces) an infinite seed-bank turns clustering into coexistence. We will also explore the impact of allowing resampling that is controlled by a random environment. Along the way we discuss robustness, universality and critical dimension.
267
290
1
10.4171/ECR/17-1/13
https://www.ems-ph.org/doi/10.4171/ECR/17-1/13
Ancestral lineages in spatial population models with local regulation
Matthias
Birkner
Johannes-Gutenberg-Universität Mainz, Germany
Nina
Gantert
TU München, Garching, Germany
We give a short overview on our work on ancestral lineages in spatial population models with local regulation. We explain how an ancestral lineage can be interpreted as a random walk in a dynamic random environment. Defining regeneration times allows to prove central limit theorems for such walks. We also consider several ancestral lineages in the same population and show for one prototypical example that in one dimension the corresponding system of coalescing walks converges to the Brownian web.
291
310
1
10.4171/ECR/17-1/14
https://www.ems-ph.org/doi/10.4171/ECR/17-1/14
The symbiotic branching model: Duality and interfaces
Jochen
Blath
TU Berlin, Germany
Marcel
Ortgiese
University of Bath, UK
The symbiotic branching model describes the dynamics of a spatial two-type popula- tion, where locally particles branch at a rate given by the frequency of the other type combined with nearest-neighbour migration. This model generalises various classic models in population dynamics, such as the stepping stone model and the mutually catalytic branching model. We are particularly interested in understanding the region of coexistence, i.e. the interface between the two types. In this chapter, we give an overview over our results that describe the dynamics of these interfaces at large scales. One of the reasons that this system is tractable is that it exhibits a rich duality theory. So at the same time, we take the opportunity to provide an introduction to the strength of duality methods in the context of spatial population models.
311
336
1
10.4171/ECR/17-1/15
https://www.ems-ph.org/doi/10.4171/ECR/17-1/15
Multitype branching models with state-dependent mutation and competition in the context of phylodynamic patterns
Anja
Sturm
Georg-August-Universität Göttingen, Germany
Anita
Winter
Universität Duisburg-Essen, Germany
In this article we propose a type-dependent branching model with mutation and competition, which can be used to model phylodynamic patterns of a virus population. For any two virus particles, the competition kernel depends on the particles’ types and the total mass of the population. We introduce our individual-based model as a measure-valued process and discuss possible scaling regimes. We then model the evolving phylogenies as stochastic processes with values in the space of marked metric measure spaces. For that we rely on the genetic information given through the number of nucleotide substitutions separating the virus particles. Finally, we construct a two- level branching model that describes branching with competition within splitting cells. In all cases the large population limit of these models solves a martingale problem. Uniqueness of solutions is verified for the first and third model. The techniques for showing uniqueness include novel formulations of Girsanov’s theorem in the presence of jumps as well as a novel duality relation in the two-level-model. In the second model uniqueness is open but conjectured to hold under restricted conditions on the branching and competition rates. For this model the proof of existence of the limit relies on a new method of showing the compact containment condition for stochastic processes with values in metric measure spaces that are a priori not ultra-metric.
337
364
1
10.4171/ECR/17-1/16
https://www.ems-ph.org/doi/10.4171/ECR/17-1/16
Ancestral lines under recombination
Ellen
Baake
Universität Bielefeld, Germany
Michael
Baake
Universität Bielefeld, Germany
Solving the recombination equation has been a long-standing challenge of deterministic population genetics. We review recent progress obtained by introducing ancestral processes, as traditionally used in the context of stochastic models of population genetics, into the deterministic setting. With the help of an ancestral partitioning process, which is obtained by letting population size tend to infinity (without rescaling parameters or time) in an ancestral recombination graph, we obtain the solution to the recombination equation in a transparent form.
365
382
1
10.4171/ECR/17-1/17
https://www.ems-ph.org/doi/10.4171/ECR/17-1/17
Towards more realistic models of genomes in populations: The Markov-modulated sequentially Markov coalescent
Julien
Dutheil
MPI für Evolutionsbiologie, Plön, Germany
The development of coalescent theory paved the way to statistical inference from population genetic data. In the genomic era, however, coalescent models are limited due to the complexity of the underlying ancestral recombination graph. The sequentially Markov coalescent (SMC) is a heuristic that enables the modelling of complete genomes under the coalescent framework. While it empowers the inference of detailed demographic history of a population from as few as one diploid genome, current implementations of the SMC make unrealistic assumptions about the homogeneity of the coalescent process along the genome, ignoring the intrinsic spatial variability of parameters such as the recombination rate. Here, I review the historical developments of SMC models and discuss the evidence for parameter heterogeneity. I then survey approaches to handle this heterogeneity, focusing on a recently developed extension of the SMC.
383
408
1
10.4171/ECR/17-1/18
https://www.ems-ph.org/doi/10.4171/ECR/17-1/18
Diffusion limits of genealogies under various modes of selection
Martin
Hutzenthaler
Universität Duisburg-Essen, Germany
Peter
Pfaffelhuber
Albert-Ludwigs-Universität Freiburg, Germany
We are studying genealogies in population genetic models including selection. Our main tool is the tree-valued Fleming–Viot process as introduced in [2]. We review approximations on the change in tree-length relative to neutrality, as well as tree- valued processes in models with fluctuating selection. The latter is treated by using an approach on stochastic averaging, which works on both time-discrete and time- continuous stochastic processes.
409
426
1
10.4171/ECR/17-1/19
https://www.ems-ph.org/doi/10.4171/ECR/17-1/19
Counting, grafting and evolving binary trees
Thomas
Wiehe
Universität zu Köln, Germany
Binary trees are fundamental objects in models of evolutionary biology and population genetics. Here, we discuss some of their combinatorial and structural properties as they depend on the tree class considered. Furthermore, the process by which trees are generated determines the probability distribution in tree space. Yule trees, for instance, are generated by a pure birth process. When considered as unordered, they have neither a closed-form enumeration nor a simple probability distribution. But their ordered siblings have both. They present the object of choice when studying tree structure in the framework of evolving genealogies.
427
450
1
10.4171/ECR/17-1/20
https://www.ems-ph.org/doi/10.4171/ECR/17-1/20
Algebraic measure trees: Statistics based on sample subtree shapes and sample subtree masses
Anita
Winter
Universität Duisburg-Essen, Germany
Null models of binary genealogical or phylogenetic trees are useful for testing hypo- theses. In this chapter we describe the space of algebraic measure trees whose elements represent phylogenies and genealogies as binary trees without edge lengths endowed with a sampling measure. With the aim of describing the degree of similarity between actual and simulated phylogenies or genealogies, we focus on the sample shape of subtrees and related statistics. We describe certain statistics of the branching and coalescent tree in more detail. Finally, we use the martingale problem method to characterise evolving trees analytically.
451
476
1
10.4171/ECR/17-1/21
https://www.ems-ph.org/doi/10.4171/ECR/17-1/21
Lectures on Selected Topics in von Neumann Algebras
Fumio
Hiai
Tohoku University, Japan
Functional analysis
Primary 46L10; secondary 46L51
Functional analysis
von Neumann algebra, Tomita–Takesaki theory, modular operator, standard form, Connes’ cocycle derivative, operator-valued weight, relative modular operator, crossed product, KMS condition, Takesaki’s duality theorem,
The theory of von Neumann algebras, originating with the work of F. J. Murray and J. von Neumann in the late 1930s, has grown into a rich discipline with connections to different branches of mathematics and physics. Following the breakthrough of Tomita–Takesaki theory, many great advances were made throughout the 1970s by H. Araki, A. Connes, U. Haagerup, M. Takesaki and others. These lecture notes aim to present a fast-track study of some important topics in classical parts of von Neumann algebra theory that were developed in the 1970s. Starting with Tomita–Takesaki theory, this book covers topics such as the standard form, Connes’ cocycle derivatives, operator-valued weights, type III structure theory and non-commutative integration theory. The self-contained presentation of the material makes this book useful not only to graduate students and researchers who want to know the fundamentals of von Neumann algebras, but also to interested undergraduates who have a basic knowledge of functional analysis and measure theory.
4
1
2021
978-3-98547-004-4
978-3-98547-504-9
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/ELM/32
https://www.ems-ph.org/doi/10.4171/ELM/32
EMS Series of Lectures in Mathematics
2523-5176
2523-5184
Topology and Geometry
A Collection of Essays Dedicated to Vladimir G. Turaev
Athanase
Papadopoulos
Université de Strasbourg and CNRS, France
History and biography
Combinatorics
Associative rings and algebras
Nonassociative rings and algebras
01-02, 05E15, 16T05, 17B37, 17B63, 18M05, 18D10, 20D60, 53D17, 57K31, 55N33, 57M27, 57N05, 68R15, 57R19, 57R56, 58D19
Mathematics and science
higher linking numbers for links in the 3-sphere, intersection of loops on surfaces, Turaev cobracket, Poincaré complexes, Poincaré duality, spin structures in 3-manifolds, explicit constructions of cocycles, Turaev surface, Turaev volume, skein module, quantum invariants of knots, links and 3-manifolds, 6j-symbols; Turaev–Viro invariant, Reshetikhin–Turaev invariant, TQFT, HQFT, Gauss words and links, enumeration problems in topology and group theory, generalizations of the Thurston norm, knotoids, knots, links, braids, intersections and self-intersections of loops on surfaces, cobrackets, combinatorial group theory, metric geometry, phylogenetics
The present volume consists of a collection of essays dedicated to Vladimir Turaev. The essays cover the large spectrum of topics in which Turaev has been interested, including knot and link invariants, quantum representations, TQFTs, state sum constructions, geometric structures on knot complements, Kleinian groups, geometric group theory and its relationship with 3-manifolds, mapping class groups, operads, mathematical physics, Grothendieck’s program, the philosophy of mathematics, and several other topics. At the same time, this volume will give an overview of topics that are at the forefront of current research in topology and geometry. Some of the essays are research articles and contain new results, sometimes answering questions that were raised by Turaev. The rest of the essays are surveys that will introduce the reader to some key ideas in the field.
7
15
2021
978-3-98547-001-3
978-3-98547-501-8
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/IRMA/33
https://www.ems-ph.org/doi/10.4171/IRMA/33
IRMA Lectures in Mathematics and Theoretical Physics
2523-5133
2523-5141
33
Vladimir Turaev, friend and colleague
Athanase
Papadopoulos
Université de Strasbourg et CNRS, France
This is a biography and a report on the work of Vladimir Turaev. Using fundamental techniques that are rooted in classical topology, Turaev introduced new ideas and tools that transformed the field of knots and links and invariants of 3-manifolds. He is one of the main founders of the new topic called quantum topology. In surveying Turaev’s work, this article will give at the same time an overview of an important part of the intense activity in low-dimensional topology that took place over the last 45 years, with its connections with mathematical physics.
15
44
1
10.4171/IRMA/33-1/1
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/1
Triangles on planar Jordan $C^1$-curves and differential topology
Jean-Claude
Hausmann
Université de Genève, Switzerland
We prove that a Jordan $\mathcal{C}^1$-curve in the plane contains the vertices of any non-flat triangle, up to translation and homothety with positive ratio. This is false if the curve is not $C1$. The proof makes use of configuration spaces, differential and algebraic topology as well as the smooth Schoenflies theorem. A partial generalization holds true in higher dimensions.
45
58
1
10.4171/IRMA/33-1/2
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/2
Null-homologous unknottings
Charles
Livingston
Indiana University, Bloomington, USA
Every knot can be unknotted with two generalized twists; this was first proved by Ohyama. Here we prove that any knot of genus g can be unknotted with 2$g$ null-homologous twists and that there exist genus $g$ knots that cannot be unknotted with fewer than 2$g$ null-homologous twists.
59
68
1
10.4171/IRMA/33-1/3
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/3
A question of Turaev about triple higher Milnor linking numbers of divide links
Norbert
A’Campo
Universität Basel, Switzerland
The following question was pronounced, many years ago in Strasbourg, by Vladimir Turaev in the IRMA lecture room: Let $L \subset S^3$ be a link consisting of divide knots $K_1;K_2; \dots ;K_n$ given by divides $P_1; P_2; \dots ;P_n$. How to compute the Milnor higher linking numbers from the system of divides? Also in the room was the question: Let $L \subset S^3$ be a link consisting of knots $K_1;K_2; \dots ;K_n$ given by Turaev shadows $S_1; S_2; \dots ;S_n$. How to compute the Milnor higher linking numbers from the system of shadows? The present contribution answers the first question for the triple higher linking numbers.
69
74
1
10.4171/IRMA/33-1/4
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/4
Fundamental groups in projective knot theory
Julia
Viro
Stony Brook University, USA
Oleg
Viro
Stony Brook University, USA
We relate properties of a link $L$ in the projective space $\mathbb{R}P^3$ to properties of the group $\pi_1(\mathbb{R}P^3 \smallsetminus L)$: \begin{itemize} \item $L$ is isotopic to a projective line if and only if $\pi_1(\mathbb{R}P^3 \smallsetminus L) = \mathbb{Z}$. \item $L$ is isotopic to an affine circle if and only if $\pi_1(\mathbb{R}P^3 \smallsetminus L) = \mathbb{Z} \ast \mathbb{Z}_{/2}$. \item $L$ is isotopic to a link disjoint from a projective plane if and only if $\pi_1(\mathbb{R}P^3 \smallsetminus L)$ contains a non-trivial element of order two. \end{itemize} A simple algorithm which finds a system of generators and relations for $\pi_1(\mathbb{R}P^3 \smallsetminus L)$ in terms of a link diagram of $L$ is provided.
75
92
1
10.4171/IRMA/33-1/5
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/5
The Affine Index Polynomial and the Sawollek Polynomial
Louis
Kauffman
University of Illinois at Chicago, USA
This chapter gives a concise proof of a relationship between the Affine Index Polynomial and the Generalized Alexander Polynomial, known as the Sawollek Polynomial. The paper is dedicated to Vladimir Turaev and to his continued creative contribution to Mathematics!
93
108
1
10.4171/IRMA/33-1/6
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/6
Introduction to quantum representations of mapping class groups
Julien
Marché
Université Paris Diderot, Sorbonne Paris Cité, France
We provide an (almost) self-contained construction of the Witten–Reshetikhin–Turaev representations of the mapping class group. We describe its properties including its Hermitian structure, irreducibility and integrality (at prime level). The construction in these notes relies only on skein theory (Kauffman Bracket) and does not use surgery techniques. We hope that they will be accessible to non-specialists.
109
130
1
10.4171/IRMA/33-1/7
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/7
On symmetric matrices associated with oriented link diagrams
Rinat
Kashaev
Université de Genève, Switzerland
Let $D$ be an oriented link diagram with the set of regions $\operatorname{r}_{D}$. We define a symmetric map (or matrix) $\tau_D$ : $\operatorname{r}_{D}$$\times$ $\operatorname{r}_{D}$ $\to$ $\mathbb{Z}{[x]}$ that gives rise to an invariant of oriented links, based on a slightly modified S-equivalence of Trotter and Murasugi in the space of symmetric matrices. In particular, for real $x$, the negative signature of $\tau_D$ corrected by the writhe is conjecturally twice the Tristram– Levine signature function, where $2x = \sqrt{t} + \frac1{\sqrt{t}}$ with $t$ being the indeterminate of the Alexander polynomial.
131
146
1
10.4171/IRMA/33-1/8
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/8
On Chen–Yang’s volume conjecture for various quantum invariants
Jun
Murakami
Waseda University, Tokyo, Japan
Chen and Yang extended the volume conjecture of quantum ${\mathcal U}_q(sl_2)$ invariant by putting $q$ = exp$(4\pi i/r)$ instead of exp$(2\pi i/r)$/ for odd $r$ and they fond that the conjecture holds not only for complements of knots but also for various hyperbolic three-manifolds. Here we show numerical verifications of several variations of their conjecture.
147
160
1
10.4171/IRMA/33-1/9
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/9
Non-semisimple invariants and Habiro’s series
Anna
Beliakova
Universität Zürich, Switzerland
Kazuhiro
Hikami
Kyushu University, Fukuoka, Japan
In this chapter we establish an explicit relationship between Habiro’s cyclotomic expansion of the colored Jones polynomial (evaluated at a $p$th root of unity) and the Akutsu–Deguchi– Ohtsuki (ADO) invariants of the double twist knots. This allows us to compare the Witten–Reshetikhin– Turaev (WRT) and Costantino–Geer–Patureau (CGP) invariants of 3-manifolds obtained by 0-surgery on these knots. The difference between them is determined by the $p - 1$ coefficient of the Habiro series. We expect these results to hold for all Seifert genus 1 knots.
161
174
1
10.4171/IRMA/33-1/10
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/10
Modular categories and TQFTs beyond semisimplicity
Christian
Blanchet
Sorbonne Université, Paris, France
Marco
De Renzi
Universität Zürich, Switzerland
Vladimir Turaev discovered in the early years of quantum topology that the notion of modular category was an appropriate structure for building 3-dimensional Topological Quantum Field Theories (TQFTs for short) containing invariants of links in 3-manifolds such as Witten– Reshetikhin–Turaev ones. In recent years, generalized notions of modular categories, which relax the semisimplicity requirement, have been successfully used to extend Turaev’s construction to various non-semisimple settings. We report on these recent developments in the domain, showing the richness of Vladimir’s lineage.
175
208
1
10.4171/IRMA/33-1/11
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/11
State sums for some super quantum link invariants
Louis-Hadrien
Robert
Université du Luxembourg, Esch-sur-Alzette, Luxembourg
Emmanuel
Wagner
Université Paris Diderot, France
We present state sums for quantum link invariants arising from the representation theory of $U_q(\mathfrak{gl}_{N \mid M})$. We investigate the case of the $N$th exterior power of the standard representation of $U_q(\mathfrak{gl}_{N \mid 1})$ and make explicit the relation with Kashaev invariants.
209
246
1
10.4171/IRMA/33-1/12
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/12
Brane Topological Field Theory and Hurwitz numbers for CW-complexes
Sergey
Natanzon
National Research University Higher School of Economics, Moscow, Russian Federation
We expand Topological Field Theory on some special CW-complexes (brane complexes). This Brane Topological Field Theory one-to-one corresponds to infinite dimensional Frobenius algebras, graded by CW-complexes of smaller dimensions. We define general and regular Hurwitz numbers of brane complexes and prove that they generate Brane Topological Field Theories. For general Hurwitz numbers, the corresponding algebra is an algebra of coverings of smaller dimension. For regular Hurwitz numbers, the Frobenius algebra is an algebra of families of subgroups of finite groups.
247
256
1
10.4171/IRMA/33-1/13
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/13
Resurgence of Faddeev’s quantum dilogarithm
Stavros
Garoufalidis
Southern University of Science and Technology, Shenzhen, China
Rinat
Kashaev
Université de Genève, Switzerland
The quantum dilogarithm function of Faddeev is a special function that plays a key role as the building block of quantum invariants of knots and 3-manifolds, of quantum Teichmüller theory and of complex Chern–Simons theory. Motivated by conjectures on resurgence and the recent interest in wall-crossing phenomena, we prove that the Borel summation of a formal power series solution of a linear difference equation produces Faddeev’s quantum dilogarithm. Along the way, we give an explicit formula for the Borel transform, a meromorphic function in the Borel plane, locate its poles and residues and describe the Stokes phenomenon of its Laplace transforms along the Stokes rays.
257
272
1
10.4171/IRMA/33-1/14
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/14
On mapping class group quotients by powers of Dehn twists and their representations
Louis
Funar
Université Grenoble Alpes, Gières, France
The aim of this chapter is to survey some known results about mapping class group quotients by powers of Dehn twists, related to their finite dimensional representations and to state some open questions. One can construct finite quotients of them, out of representations with Zariski dense images into semisimple Lie groups. We show that, in genus 2, the Fibonacci TQFT representation is actually a specialization of the Jones representation. Eventually, we explain a method of Long and Moody which provides large families of mapping class group representations.
273
308
1
10.4171/IRMA/33-1/15
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/15
Higher holonomy and iterated integrals
Toshitake
Kohno
University of Tokyo, Japan
We develop a method to construct representations of the homotopy $n$-groupoid of a manifold as an $n$-category by means of K.-T. Chen’s formal homology connections for any positive integer $n$. We establish a higher holonomy functor from the homotopy $n$-groupoid to a category obtained from the tensor algebra over the homology of the manifold.
309
328
1
10.4171/IRMA/33-1/16
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/16
Some algebraic aspects of the Turaev cobracket
Nariya
Kawazumi
University of Tokyo, Japan
The Turaev cobracket, a loop operation introduced by V. Turaev [52], which measures the self-intersection of a loop on a surface, is a modification of a path operation introduced earlier by Turaev himself [51], as well as a counterpart of the Goldman bracket [22]. In this survey based on the author’s joint works with A. Alekseev, Y. Kuno and F. Naef, we review some algebraic aspects of the cobracket and its framed variants including their formal description, an application to the mapping class group of the surface and a relation to the (higher genus) Kashiwara–Vergne problem. In addition, we review a homological description of the cobracket after R. Hain [24].
329
356
1
10.4171/IRMA/33-1/17
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/17
Generalized Dehn twists in low-dimensional topology
Yusuke
Kuno
Tsuda University, Tokyo, Japan
Gwénaël
Massuyeau
Université de Bourgogne Franche-Comté, Dijon, France
Shunsuke
Tsuji
Meiji University, Tokyo, Japan
The generalized Dehn twist along a closed curve in an oriented surface is an algebraic construction which involves intersections of loops in the surface. It is defined as an automorphism of the Malcev completion of the fundamental group of the surface. As the name suggests, for the case where the curve has no self-intersection, it is induced from the usual Dehn twist along the curve. In this expository article, after explaining their definition, we review several results about generalized Dehn twists such as their realizability as diffeomorphisms of the surface, their diagrammatic description in terms of decorated trees and the Hopf-algebraic framework underlying their construction. Going to dimension three, we also overview the relation between generalized Dehn twists and 3-dimensional homology cobordisms, and we survey the variants of generalized Dehn twists for skein algebras of the surface.
357
398
1
10.4171/IRMA/33-1/18
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/18
On geometric group theory
Valentin
Poénaru
Université Paris Sud-Orsay, France
399
432
1
10.4171/IRMA/33-1/19
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/19
Geometry of knots and links
Nikolay
Abrosimov
Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russian Federation
Alexander
Mednykh
Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russian Federation
We give an overview of recent results on the geometry of knots and links. More precisely, we investigate the existence of hyperbolic, spherical or Euclidean structure on various cone manifolds whose underlying space is the three-dimensional sphere and whose singular set is a given knot or link. We present trigonometrical identities involving the lengths of singular geodesics and cone angles of such cone manifolds. Then these identities are used to produce exact integral formulas for volumes of the corresponding cone manifolds.
433
454
1
10.4171/IRMA/33-1/20
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/20
Essential closed surfaces and finite coverings of negatively curved cusped 3-manifolds
Charalampos
Charitos
Agricultural University of Athens, Greece
The existence of essential closed surfaces is proven for finite coverings of 3-manifolds that are triangulated by finitely many topological ideal tetrahedra and admit a regular, negatively curved, ideal structure.
455
476
1
10.4171/IRMA/33-1/21
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/21
Continuous and discontinuous functions on deformation spaces of Kleinian groups
Ken’ichi
Ohshika
Gakushuin University, Tokyo, Japan
To understand the topologies of deformation spaces of Kleinian groups, it is helpful to make use of functions defined on deformation spaces and to study their continuity and discontinuity. In this chapter, we focus on the three functions, the length function, the end invariant function, and the two-variable Cannon–Thurston map, restricting our attention to deformation spaces of Kleinian surface groups, and study how they behave on the boundaries of deformation spaces. We show that although the length function is continuous on the entire deformation space, both the end invariant function and the two-variable Cannon–Thurston map have discontinuous points, reflecting the difference of geometric limits and algebraic limits. Furthermore, we show, using several illuminating examples, that the end invariant function and the two-variable Cannon–Thurston map have different sensibilities towards geometric limits.
477
502
1
10.4171/IRMA/33-1/22
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/22
A generalization of King’s equation via noncommutative geometry
Gourab
Bhattacharya
Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France
Maxim
Kontsevich
Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France
We introduce a framework in noncommutative geometry consisting of a $\ast$-algebra, a bimodule endowed with a derivation (“1-forms”) and a Hermitian structure (a “noncommutative Kähler form”), and a cyclic 1-cochain whose coboundary is determined by the previous structures. This data leads to moment map equations on the space of connections on arbitrary finitely-generated projective Hermitian module. As particular cases, we obtain a large class of equations in algebra (King’s equations for representations of quivers, including ADHM equations), in classical gauge theory (Hermitian Yang–Mills equations, Hitchin equations, Bogomolny and Nahm equations, etc.), as well as in noncommutative gauge theory by Connes, Douglas and Schwarz. We also discuss Nekrasov’s beautiful proposal for re-interpreting noncommutative instantons on $\mathbb{C}^n \simeq \mathbb{R}^{2n}$ as an infinite-dimensional solution of King’s equation $$ \displaystyle\sum_{i=1}^{n} [T_i^\dagger,T_i] = \hbar \cdot n \cdot \operatorname{id}_{\mathcal{H}} $$ where $\mathcal{H}$ is a Hilbert space completion of a finitely-generated $\mathbb{C}[T_1,\dots, T_n]$-module (e.g., an ideal of finite codimension).
503
536
1
10.4171/IRMA/33-1/23
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/23
Dessins for modular operads and the Grothendieck–Teichmüller group
Noémie C.
Combe
Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Germany
Yuri I.
Manin
Max-Planck-Institut für Mathematik, Bonn, Germany
Matilde
Marcolli
California Institute of Technology, Pasadena, USA
A part of Grothendieck’s program for studying the Galois group $G_{\mathbf{Q}}$ of the field of all algebraic numbers $\overline{\mathbf{Q}}$ emerged from his insight that one should lift its action upon $\overline{\mathbf{Q}}$ to the action of $G_{\mathbf{Q}}$ upon the (appropriately defined) profinite completion of $\pi_1(\mathbf{P}^1 \{0, 1, \infty \})$. The latter admits a good combinatorial encoding via finite graphs, “dessins d’enfant”. This part was actively developing during the last decades, starting with foundational works of A. Belyi, V. Drinfeld and Y. Ihara. This chapter concerns another part of Grothendieck’s program, in which its geometric environment is extended to moduli spaces of algebraic curves, more specifically, stable curves of genus zero with marked/labeled points. Our main goal is to show that dual graphs of such curves may play the role of “modular dessins” in an appropriate operadic context.
537
560
1
10.4171/IRMA/33-1/24
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/24
On the involution Jimm
A. Muhammed
Uludağ
Galatasaray University, Istanbul, Turkey
This chapter is a survey of the involution of the real line induced by Dyer’s outer automorphism of the group PGL(2,Z). This ‘modular’ involution is discontinuous at the rationals but satisfies a surprising collection of functional equations. It preserves the set of real quadratic irrationals mapping them in a non-obvious way to each other. It commutes with the Galois action on real quadratic irrationals. It restricts to a non-trivial involution of the set of elements of norm +1 in real quadratic number fields. More generally, it preserves set-wise the orbits of the modular group, thereby inducing an involution of the moduli space of real rank-two lattices. We give a description of this involution as the boundary action of a certain automorphism of the infinite trivalent tree. It is conjectured that algebraic numbers of degree at least three are mapped to transcendental numbers under this involution.
561
578
1
10.4171/IRMA/33-1/25
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/25
What is a black hole? – A geometric introduction
Sumio
Yamada
Gakushuin University, Tokyo, Japan
Black hole is a term which fascinates many for its philosophical consequences, yet puzzles those who wonder how such a thing could possibly exist. We will survey the historical passage for the theory of relativity, and try to demonstrate the inevitability of the concept of black hole, once we accept the Einstein equation and its geometric content.
579
598
1
10.4171/IRMA/33-1/26
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/26
The paradoxical nature of mathematics
Vassiliki
Farmaki
National and Kapodistrian University of Athens, Greece
Stelios
Negrepontis
National and Kapodistrian University of Athens, Greece
Mathematics is usually described as a deductive science. The set of axioms, with which we start, should be as economical as possible, hopefully consistent, and deductively strong with as many as possible “desirable” consequences. How do we achieve sufficient $deductive$ $power$ for an axiomatic system? In this work we present the unorthodox thesis that the deductive strength in Mathematics comes, perhaps exclusively, from its paradoxical nature, namely from its proximity to the contradictory, a proximity that almost always takes the form of a Finitization of the Infinite. We support our thesis by examining Euclidean Geometry; Number Theory; Incommensurability and periodic anthyphairesis/continued fractions in the Mathematics and the Philosophy of the Pythagoreans, Zeno, Theaetetus, Plato; ratios of magnitudes and method of exhaustion in weakly finitized form by Eudoxus, and Real numbers and Calculus in the strongly finitized form by Dedekind completeness; Set theory axioms such as the axiom of choice, with special reference to compactness and ultrafilters, and Gödel’s program with axioms of large cardinals. In the last two sections we argue that Beauty in Mathematics and we suggest that the “Unreasonable Effectiveness of Mathematics in the Natural Sciences” are both manifestations of the Paradoxical Nature of Mathematics.
599
642
1
10.4171/IRMA/33-1/27
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/27
Returns of geometry: From the Pythagoreans to mathematical modernism and beyond
Arkady
Plotnitsky
Purdue University, West Lafayette, USA
This article considers the persistence of geometrical thinking in mathematics amidst the twentieth-century transformation of mathematics associated with “mathematical modernism,” which may be characterized, as it was by this author previously, by an algebraization of mathematics, even in such fields as geometry and topology. This article, by contrast, discusses the role of geometry or, more generally (so as to include topology and related fields), the mathematics of spatiality, its “returns” to the scene of mathematics, in its interaction with algebra during this period. Thus, it also argues for a continuity, along with a break, between mathematical modernism with the preceding history of mathematics. As part of this history, I shall consider a form of thinking, in mathematics and beyond, defined by the role of the unthinkable in thought, or in the language of the ancient Greeks, who introduced this architecture of thought with the discovery of the incommensurable magnitudes in Pythagorean mathematics, the alogon within a logos. I shall argue that this architecture, which takes a more radical form, designated here as radical Pythagorean thinking (in mathematics itself and in physics), underlies the interplay of algebra and geometry in mathematical modernism and beyond.
643
682
1
10.4171/IRMA/33-1/28
https://www.ems-ph.org/doi/10.4171/IRMA/33-1/28
Geometry and Topology of Surfaces
Sebastian
Baader
Universität Bern, Switzerland
Manifolds and cell complexes
57K20
Mathematics and science
Mathematics
mapping class group, Dehn twist, pseudo-Anosov diffeomorphism, hyperbolic surface, Basmajian identity, measured foliation, Teichmüller theory, Thurston classification
These lecture notes cover the classification of hyperbolic structures and measured foliations on surfaces in a minimalist way. While the inspiration is obviously taken from the excellent books Primer on mapping class groups and Travaux de Thurston sur les surfaces, the author aimed at including a little bit more of hyperbolic trigonometry, including a proof of Basmajian's identity on the orthogeodesic spectrum, while keeping the rest short.
3
31
2021
978-3-98547-000-6
978-3-98547-500-1
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/ZLAM/26
https://www.ems-ph.org/doi/10.4171/ZLAM/26
Zurich Lectures in Advanced Mathematics
Quasi-Periodic Solutions of Nonlinear Wave Equations on the $d$-Dimensional Torus
Massimiliano
Berti
SISSA, Trieste, Italy
Philippe
Bolle
Avignon Université, France
Dynamical systems and ergodic theory
Partial differential equations
37K55, 37K50, 35L05; 35Q55
Calculus + mathematical analysis
Infinite-dimensional Hamiltonian systems, nonlinear wave equation, KAM for PDEs, quasi-periodic solutions and invariant tori, small divisors, Nash–Moser theory, multiscale analysis
Many partial differential equations (PDEs) arising in physics, such as the nonlinear wave equation and the Schrödinger equation, can be viewed as infinite-dimensional Hamiltonian systems. In the last thirty years, several existence results of time quasi-periodic solutions have been proved adopting a "dynamical systems" point of view. Most of them deal with equations in one space dimension, whereas for multidimensional PDEs a satisfactory picture is still under construction. An updated introduction to the now rich subject of KAM theory for PDEs is provided in the first part of this research monograph. We then focus on the nonlinear wave equation, endowed with periodic boundary conditions. The main result of the monograph proves the bifurcation of small amplitude finite-dimensional invariant tori for this equation, in any space dimension. This is a difficult small divisor problem due to complex resonance phenomena between the normal mode frequencies of oscillations. The proof requires various mathematical methods, ranging from Nash–Moser and KAM theory to reduction techniques in Hamiltonian dynamics and multiscale analysis for quasi-periodic linear operators, which are presented in a systematic and self-contained way. Some of the techniques introduced in this monograph have deep connections with those used in Anderson localization theory. This book will be useful to researchers who are interested in small divisor problems, particularly in the setting of Hamiltonian PDEs, and who wish to get acquainted with recent developments in the field.
10
15
2020
978-3-03719-211-5
978-3-03719-711-0
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/211
https://www.ems-ph.org/doi/10.4171/211
EMS Monographs in Mathematics
2523-5192
2523-5206
Classification of Complex Algebraic Surfaces
Ciro
Ciliberto
Università di Roma Tor Vergata, Italy
Algebraic geometry
Primary: 14J26, 14J27, 14J28, 14J29, 14E05, 14E07, 14E30; secondary: 14N05, 14J10
Algebraic geometry
Algebraic surfaces, classification
Τhe classification of complex algebraic surfaces is a very classical subject which goes back to the old Italian school of algebraic geometry with Enriques and Castelnuovo. However, the exposition in the present book is modern and follows Mori's approach to the classification of algebraic varieties. The text includes the $P_{12}$ theorem, the Sarkisov programme in the surface case and the Noether–Castelnuovo theorem in its classical version. This book serves as a relatively quick and handy introduction to the theory of algebraic surfaces and is intended for readers with a good knowledge of basic algebraic geometry. Although an acquaintance with the basic parts of books like Principles of Algebraic Geometry by Griffiths and Harris or Algebraic Geometry by Hartshorne should be sufficient, the author strove to make the text as self-contained as possible and, for this reason, a first chapter is devoted to a quick exposition of some preliminaries.
6
30
2020
978-3-03719-210-8
978-3-03719-710-3
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/210
https://www.ems-ph.org/doi/10.4171/210
EMS Series of Lectures in Mathematics
2523-5176
2523-5184
Mackey 2-Functors and Mackey 2-Motives
Paul
Balmer
University of California, Los Angeles, USA
Ivo
Dell'Ambrogio
Université de Lille, France
Group theory and generalizations
Category theory; homological algebra
$K$-theory
Algebraic topology
Primary: 20J05, 18B40, 55P91; secondary: 18M30, 18N10, 18N25, 18N40, 19A22, 20C99
Mathematics and science
Groupoids, Mackey formula, equivariant, 2-functors, derivators, ambidexterity, separable monadicity, spans, string diagrams, motivic decompositions, Burnside algebras
This book is dedicated to equivariant mathematics, specifically the study of additive categories of objects with actions of finite groups. The framework of Mackey 2-functors axiomatizes the variance of such categories as a function of the group. In other words, it provides a categorification of the widely used notion of Mackey functor, familiar to representation theorists and topologists. The book contains an extended catalogue of examples of such Mackey 2-functors that are already in use in many mathematical fields from algebra to topology, from geometry to KK-theory. Among the first results of the theory, the ambidexterity theorem gives a way to construct further examples and the separable monadicity theorem explains how the value of a Mackey 2-functor at a subgroup can be carved out of the value at a larger group, by a construction that generalizes ordinary localization in the same way that the étale topology generalizes the Zariski topology. The second part of the book provides a motivic approach to Mackey 2-functors, 2-categorifying the well-known span construction of Dress and Lindner. This motivic theory culminates with the following application: The idempotents of Yoshida’s crossed Burnside ring are the universal source of block decompositions. The book is self-contained, with appendices providing extensive background and terminology. It is written for graduate students and more advanced researchers interested in category theory, representation theory and topology.
7
31
2020
978-3-03719-209-2
978-3-03719-709-7
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/209
https://www.ems-ph.org/doi/10.4171/209
EMS Monographs in Mathematics
2523-5192
2523-5206
$K3$ Surfaces
Shigeyuki
Kondō
Nagoya University, Japan
Algebraic geometry
Several complex variables and analytic spaces
Primary 14J28; secondary 14C34, 14J10, 14J15, 14J50, 32G20
Analytic geometry
$K3$ surface, Enriques surface, Kummer surface, Torelli-type theorem, period, lattice, reflection group, automorphism group
$K3$ surfaces are a key piece in the classification of complex analytic or algebraic surfaces. The term was coined by A. Weil in 1958 – a result of the initials Kummer, Kähler, Kodaira, and the mountain K2 found in Karakoram. The most famous example is the Kummer surface discovered in the 19th century. $K3$ surfaces can be considered as a 2-dimensional analogue of an elliptic curve, and the theory of periods – called the Torelli-type theorem for $K3$ surfaces – was established around 1970. Since then, several pieces of research on $K3$ surfaces have been undertaken and more recently $K3$ surfaces have even become of interest in theoretical physics. The main purpose of this book is an introduction to the Torelli-type theorem for complex analytic $K3$ surfaces, and its applications. The theory of lattices and their reflection groups is necessary to study $K3$ surfaces, and this book introduces these notions. The book contains, as well as lattices and reflection groups, the classification of complex analytic surfaces, the Torelli-type theorem, the subjectivity of the period map, Enriques surfaces, an application to the moduli space of plane quartics, finite automorphisms of $K3$ surfaces, Niemeier lattices and the Mathieu group, the automorphism group of Kummer surfaces and the Leech lattice. The author seeks to demonstrate the interplay between several sorts of mathematics and hopes the book will prove helpful to researchers in algebraic geometry and related areas, and to graduate students with a basic grounding in algebraic geometry.
3
31
2020
978-3-03719-208-5
978-3-03719-708-0
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/208
https://www.ems-ph.org/doi/10.4171/208
EMS Tracts in Mathematics
32
Decision Support Systems for Water Supply Systems
Smart Water System to Improve the Operation of Water Supply Systems by Using Applied Mathematics
Andreas
Pirsing
Siemens AG, Berlin, Germany
Antonio
Morsi
Universität Erlangen-Nürnberg, Germany
Computer science
Fluid mechanics
Operations research, mathematical programming
Systems theory; control
68U07, 76B75, 93C05, 90C11, 90C30, 65L80, 90-08, 90-10, 90-11
Applied mathematics
Computing and information technology
Optimization, simulation, modelling, automation, ICT, decision support system, digital twin, water supply, digitalization, automation, energy management
The book summarizes the results of the BMBF funded joint research project EWave (reference 02WER1323F) that was initiated to develop an innovative Decision Support Systems (DSS) for water supply companies. The book is written for automation experts in water supply companies as well as mathematicians who work for infrastructure companies. Operating water supply systems is complex. It has to be ensured that consumers are reliably supplied with a sufficient quantity and quality of water as well as a sufficient water pressure at all times. In addition to a reliable water supply, consumers demand for reasonable prices. For decision making and operational support, the EWave system uses newly developed integrated optimization modules. As a result, the user receives operating schedules on a 15 minute scale. For this purpose, mixed-integer linear and nonlinear mathematical optimization methods are combined. First, a mixed-integer optimization model is solved in order to derive all discrete decisions (primarily pump schedules). The idea here is to approximate the physics by piecewise linear relaxations well enough to come up with the right/optimal decisions. EWave then uses nonlinear optimization and simulation methods to get the physics straight. The process is iterated if necessary. This approach enables globally optimal solutions within an a-priori given quality tolerance. Optimization results obtained in real time yield a potential of energy savings of up to 4–6% daily for the waterworks in the pilot area.
5
31
2020
978-3-03719-207-8
978-3-03719-707-3
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/207
https://www.ems-ph.org/doi/10.4171/207
EMS Series in Industrial and Applied Mathematics
2523-5087
2523-5095
2
Role of ICT in water supply systems: requirements, current status and challenges
Andreas
Pirsing
Siemens AG, Berlin, Germany
Moritz
Allmaras
Siemens AG, München, Germany
Roland
Rosen
Siemens AG, München, Germany
Tim
Schenk
Siemens AG, München, Germany
Annelie
Sohr
Siemens AG, München, Germany
Computer science
The advances in information and communication technology (ICT) have already led to new technological solutions in the water industry. This continues a trend that started long time ago, but in many cases has been limited by several technical restrictions. Digitalization is also changing economic activities in many ways: workflows and forms of organization are being transformed to the emergence of new business models. This chapter describes the ICT basics that are relevant for the water industry and lays the theoretical foundations for understanding the newly developed decision support system EWave.
3
17
1
10.4171/207-1/1
https://www.ems-ph.org/doi/10.4171/207-1/1
EWave energy management system
Constantin
Blanck
RWW Rheinisch-Westfälische Wasserwerksgesellschaft mbH, Mülheim a.d. Ruhr, Germany
Stefan
Fischer
Netzgesellschaft Düsseldorf mbH, Germany
Michael
Plath
RWW Rheinisch-Westfälische Wasserwerksgesellschaft mbH, Mülheim a.d. Ruhr, Germany
Moritz
Allmaras
Siemens AG, München, Germany
Andreas
Pirsing
Siemens AG, Berlin, Germany
Tim
Schenk
Siemens AG, München, Germany
Annelie
Sohr
Siemens AG, München, Germany
Computer science
The EWave energy management system is a decision support system which gives support for an energy and cost efficient operation of water infrastructures. The primary objectives of water supply, which are security of supply and water quality, are extended to include also energy efficiency and automation aspects. The Dorsten-Holsterhausen pilot network provides optimal conditions for testing and evaluating the EWave system in the Rheinisch-Westfälische Wasserwerksgesellschaft mbH (RWW) distribution network. The conditions from the real pilot system to be met during operation and the potentials for improvement to be exploited are translated into various requirements for the development of the EWave system concerning architecture, data interfaces, computational modules and system models.
19
35
1
10.4171/207-1/2
https://www.ems-ph.org/doi/10.4171/207-1/2
Demand forecast
Patrick
Hausmann
Hochschule Bonn-Rhein-Sieg, Sankt Augustin, Germany
Computer science
The following chapter deals with the statistical demand forecast for the waterworks in Dorsten-Holsterhausen, which was developed during the EWave research project. Basic principle for the forecast development was the assumption that demand levels on similar days (e.g. every Monday in February) are equal. Therefore the historical demand levels were analysed and classified depending on time, day and year (clustering). The historical data was provided by the water supply company RWW as hourly measured time series values from a period of several years. After the analysis, demand changes were fitted with first order polynomials (regression lines). Thereby forecast values with a time resolution of one hour could be calculated. To avoid discrete transitions between forecast values and increase their time resolution a spline interpolation was carried out afterwards. It allowed artificial time resolutions of up to one second and guaranteed continuously differentiable forecast values. In the end forecast results were compared with current measures at RWW to determine and analyse accuracy and quality of the forecast program.
39
49
1
10.4171/207-1/3
https://www.ems-ph.org/doi/10.4171/207-1/3
Hydraulic modeling and energy view
Gerd
Steinebach
Hochschule Bonn-Rhein-Sieg, Sankt Augustin, Germany
Oliver
Kolb
Universität Mannheim, Germany
Computer science
In this chapter the hydraulic modeling of a waterworks and the drinking water distribution network is considered. At first, network elements are introduced, which allow the mapping of the real water supply network to the mathematical model. For each network element a mathematical description is given and coupling conditions are defined. Moreover, each element can be provided with an elctrical power requirement, in order to compute the overall power demand for the whole network. For the resulting mathematical model two simulators are introduced. TWaveSim can only be used for simulation, while Anaconda is also suitable for continuous optimization. On the other hand, TWaveSim is more accurate for the simulation purpose and must be applied first, to compute initial values that are required for Anaconda. Finally, a test example is constructed and simulated. This example includes most of the introduced network elements and is used to show all mathematical equations in detail and to compare the two simulators.
51
71
1
10.4171/207-1/4
https://www.ems-ph.org/doi/10.4171/207-1/4
Optimization
Björn
Geißler
Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
Alexander
Martin
Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
Antonio
Morsi
Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
Maximilian
Walther
Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
Oliver
Kolb
Universität Mannheim, Germany
Jens
Lang
Technische Hochschule Darmstadt, Germany
Lisa
Wagner
Technische Hochschule Darmstadt, Germany
Computer science
The core component of EWave is constituted by a receding horizon optimal control algorithm that is implemented in the EWave optimization module (EWave-OPT). EWave-OPT is made up of two major components, the discrete optimization module, which is responsible for computing optimal discrete switching decisions on the basis of a quasi-stationary approximation of the physical reality, and the continuous optimization module (EWave-NOPT) that, in a second step, computes optimal values for the continuous control variables subject to fixed discrete controls and an instationary, highly accurate pipeflow model.
73
104
1
10.4171/207-1/5
https://www.ems-ph.org/doi/10.4171/207-1/5
Network aggregation
Tim
Jax
Hochschule Bonn-Rhein-Sieg, Sankt Augustin, Germany
Computer science
Simulating real network often proves to be challenging due to the huge number of elements given. For this reason, aggregation strategies are recommended in order to reduce a network’s dimensions and, thus, computational efforts. This section introduces corresponding strategies used to create appropriate network models with respect to the EWave project. In this context, a novel approach will be discussed that – particularly based on manual processes – realizes network aggregations with a self-defined extent of pipe reduction and layout flexibility.
107
127
1
10.4171/207-1/6
https://www.ems-ph.org/doi/10.4171/207-1/6
Setup of simulation model and calibration
Gerd
Steinebach
Hochschule Bonn-Rhein-Sieg, Sankt Augustin, Germany
David
Dreistadt
Hochschule Bonn-Rhein-Sieg, Sankt Augustin, Germany
Patrick
Hausmann
Hochschule Bonn-Rhein-Sieg, Sankt Augustin, Germany
Tim
Jax
Hochschule Bonn-Rhein-Sieg, Sankt Augustin, Germany
Computer science
This chapter deals with the setup of the simulation model of pressure zone Holsterhausen. This model consists of two parts: The relevant processes within the waterworks Dorsten-Holsterhausen and the distribution network for drinking water. These parts are connected by the drinking water pumps. To consider the huge distribution network within the simulation model an abstraction is required, leading to arregated pipes and tanks. The final simulation model is calibrated. Calibration is made separately for single network elements like pumps and valves and for the aggregated network. The calibration of e.g. pumps can be done automatically, whereas the calibration of the network is largely a manual process. Finally, some typical simulation results are discussed. The achieved accuracy is appropriate for practical application and the further optimization process.
129
146
1
10.4171/207-1/7
https://www.ems-ph.org/doi/10.4171/207-1/7
Field data, automation, instrumentation and communication
Constantin
Blanck
RWW Rheinisch-Westfälische Wasserwerksgesellschaft mbH, Mülheim a.d. Ruhr, Germany
Stefan
Fischer
Netzgesellschaft Düsseldorf mbH, Germany
Michael
Plath
RWW Rheinisch-Westfälische Wasserwerksgesellschaft mbH, Mülheim a.d. Ruhr, Germany
Computer science
For the analysis, existing data were first digitized and made available to the project partners for model construction. The system data in the waterworks as well as the data of the pipe network are to be named here. Measurements were retrofitted at various points in the waterworks and in the pipe network. The recorded operating data was then made available to project partners for the calibration of the models and for test runs of the EWave system.
147
149
1
10.4171/207-1/8
https://www.ems-ph.org/doi/10.4171/207-1/8
New ICT architecture
Tim
Schenk
Siemens AG, München, Germany
Moritz
Allmaras
Siemens AG, München, Germany
Andreas
Pirsing
Siemens AG, Berlin, Germany
Annelie
Sohr
Siemens AG, München, Germany
Computer science
In this chapter, we discuss which indicators and views of water supply systems are needed to address all relevant user roles targeted by a decision support system. We begin with an overview of the indicators that are used to estimate and compare the efficiency of water supply systems. The specific energy consumption is only of limited usefulness in this context, hence hydraulic and electric efficiency indicators are defined that allow the formulation of an overall efficiency of a plant. Then, the relevant user roles and related quality attributes for a decision support system are investigated. The chapter concludes with a detailed discussion on how the user interface for the EWave system has been designed and implemented in order to meet these system qualities.
151
182
1
10.4171/207-1/9
https://www.ems-ph.org/doi/10.4171/207-1/9
Water cockpit: dashboards for decision support systems
Michael
Plath
RWW Rheinisch-Westfälische Wasserwerksgesellschaft mbH, Mülheim a.d. Ruhr, Germany
Constantin
Blanck
RWW Rheinisch-Westfälische Wasserwerksgesellschaft mbH, Mülheim a.d. Ruhr, Germany
Stefan
Fischer
Netzgesellschaft Düsseldorf mbH, Germany
Moritz
Allmaras
Siemens AG, München, Germany
Andreas
Pirsing
Siemens AG, Berlin, Germany
Tim
Schenk
Siemens AG, München, Germany
Annelie
Sohr
Siemens AG, München, Germany
Computer science
In this chapter, it is discussed which indicators and views of water supply systems are needed to address all relevant user roles targeted by a decision support system. The first step is an overview of the indicators that are used to estimate and compare the efficiency of water supply systems. The specific energy consumption is only of limited usefulness in this context, hence hydraulic and electric efficiency indicators are defined that allow the formulation of an overall efficiency of a plant. Then, the relevant user roles and related quality attributes for a decision support system are investigated. The chapter concludes with a detailed discussion on how the user interface for the EWave system has been designed and implemented in order to meet these system qualities.
183
198
1
10.4171/207-1/10
https://www.ems-ph.org/doi/10.4171/207-1/10
Field test
Annelie
Sohr
Siemens AG, München, Germany
Constantin
Blanck
RWW Rheinisch-Westfälische Wasserwerksgesellschaft mbH, Mülheim a.d. Ruhr, Germany
Stefan
Fischer
Netzgesellschaft Düsseldorf mbH, Germany
Michael
Plath
RWW Rheinisch-Westfälische Wasserwerksgesellschaft mbH, Mülheim a.d. Ruhr, Germany
Moritz
Allmaras
Siemens AG, München, Germany
Tim
Schenk
Siemens AG, München, Germany
Andreas
Pirsing
Siemens AG, Berlin, Germany
Computer science
The validation of the EWave system has been performed in two different ways. Main focus was the installation as pilot application directly on site at the RWW facility in the waterworks Dorsten-Holsterhausen. Before the installation and usage various IT security requirements had to be fulfilled and the control center operators were introduced to the goals and the potential of the EWave project. Then the EWave system was initially run in parallel to the real operation, without applying the setpoints on the real plant. Finally in the last stage of this test proposed setpoints were applied to the plant control. In parallel to this pilot test a further validation has been performed with the focus to quantify the benefit of EWave. To be able to assess the EWave benefit as closely as possible a concept has been developed that compared simulation results of a real historic plant operation with a theoretical EWave optimized plant operation. Finally, the results of the whole EWave project and the two validation approaches are evaluated and summarized and the potential of the approach is assessed in an outlook.
201
216
1
10.4171/207-1/11
https://www.ems-ph.org/doi/10.4171/207-1/11
Accuracy of Mathematical Models
Dimension Reduction, Homogenization, and Simplification
Sergey
Repin
Russian Acadademy of Sciences, St. Petersburg, Russian Federation
Stefan
Sauter
Universität Zürich, Switzerland
Partial differential equations
Calculus of variations and optimal control; optimization
Numerical analysis
Mechanics of deformable solids
35-02; 35J20, 35J50, 35J60, 35J88, 49M29, 65N15, 65N85, 74K20
Differential equations
Modelling error, a posteriori error majorant, model simplification, dimension reduction, homogenization, conversion of models
The expansion of scientific knowledge and the development of technology are strongly connected with quantitative analysis of mathematical models. Accuracy and reliability are the key properties we wish to understand and control. This book presents a unified approach to the analysis of accuracy of deterministic mathematical models described by variational problems and partial differential equations of elliptic type. It is based on new mathematical methods developed to estimate the distance between a solution of a boundary value problem and any function in the admissible functional class associated with the problem in question. The theory is presented for a wide class of elliptic variational problems. It is applied to the investigation of modelling errors arising in dimension reduction, homogenization, simplification, and various conversion methods (penalization, linearization, regularization, etc.). A collection of examples illustrates the performance of error estimates.
7
31
2020
978-3-03719-206-1
978-3-03719-706-6
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/206
https://www.ems-ph.org/doi/10.4171/206
EMS Tracts in Mathematics
33
Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA)
Volume 2
Frédéric
Chapoton
Université de Strasbourg, France
Frédéric
Fauvet
Université de Strasbourg, France
Claudia
Malvenuto
Università di Roma La Sapienza, Italy
Jean-Yves
Thibon
Université Paris-Est Marne-la-Vallée, France
Quantum theory
Combinatorics
Ordinary differential equations
Dynamical systems and ergodic theory
05E, 81T15, 81T18, 81Q30, 34C20, 37C10, 18D50, 34M40, 34M60, 11M32, 30D60
Quantum physics (quantum mechanics)
Operad, Hopf algebra, algebraic combinatorics, moulds, renormalization, periods, multiple zeta values, resurgent functions, alien calculus, vector fields, diffeomorphisms
This is volume 2 of a 2-volume work comprising a total of 14 refereed research articles which stem from the CARMA Conference (Algebraic Combinatorics, Resurgence, Moulds and Applications), held at the Centre International de Rencontres Mathématiques in Luminy, France, from June 26 to 30, 2017. The conference did notably emphasise the role of Hopf algebraic techniques and related concepts (e.g. Rota–Baxter algebras, operads, Ecalle’s mould calculus) which have lately proved pervasive in combinatorics, but also in many other fields, from multiple zeta values to the algebraic study of control systems and the theory of rough paths. The volumes should be useful to researchers or graduate students in mathematics working in these domains and to theoretical physicists involved with resurgent functions and alien calculus.
2
28
2020
978-3-03719-205-4
978-3-03719-705-9
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/205
https://www.ems-ph.org/doi/10.4171/205
IRMA Lectures in Mathematics and Theoretical Physics
2523-5133
2523-5141
32
Special values of finite multiple harmonic $q$-series at roots of unity
Henrik
Bachmann
Nagoya University, Japan
Yoshihiro
Takeyama
University of Tsukuba, Japan
Koji
Tasaka
Aichi Prefectural University, Nagakute-shi, Aichi, Japan
We study special values of finite multiple harmonic $q$-series at roots of unity. These objects were recently introduced by the authors and it was shown that they have connections to finite and symmetric multiple zeta values and the Kaneko–Zagier conjecture. In this note, we give new explicit evaluations for finite multiple harmonic $q$-series at roots of unity and prove Ohno–Zagier-type relations for them.
1
18
1
10.4171/205-1/1
https://www.ems-ph.org/doi/10.4171/205-1/1
Mould calculus: from primary to secondary mould symmetries
Olivier
Bouillot
Université Paris-Est Marne-la-Vallée, France
Mould calculus, moulds, formal moulds, primary mould symmetries, secondary mould symmetries, shuffle product, quasi-shuffle product, Hopf algebra, coproduct
This article deals with a part of mould calculus, a powerful combinatorial environment developed by J. Ecalle in the 80s. Its main goal is to give a complete introduction to the secondary mould symmetries, as well as to develop the path from the primary symmetries to the secondary symmetries. We first present in §2 all the classical results on moulds and prove them completely in §3. Then, we introduce the secondary symmetries in §4 and extend to them the Hopf algebraic interpretation of mould calculus. Using this, we finally give in §5 the path to go from primary to secondary symmetries.
19
84
1
10.4171/205-1/2
https://www.ems-ph.org/doi/10.4171/205-1/2
Renormalisation and locality: branched zeta values
Pierre
Clavier
Universität Potsdam, Germany
Li
Guo
Rutgers University, Newark, USA
Sylvie
Paycha
Universität Potsdam, Germany
Bin
Zhang
Sichuan University, Chengdu, China
Locality, Rota-Baxter algebra, symbols, branched zeta values
General algebraic systems
Associative rings and algebras
Several complex variables and analytic spaces
Quantum theory
Multivariate techniques are implemented in order to build, study and renormalise branched zeta functions associated with rooted trees. For this purpose, we fi rst prove algebraic results and develop analytic tools, which we then combine to study branched zeta functions. The algebraic aspects concern universal properties for locality algebraic structures; we branch/lift to trees operators on the decoration set, and factorise branched maps through words by means of universal properties for words. The analytic tools arise in the context of multivariate meromorphic germs of symbols with linear poles. The latter form a locality algebra on which we build various locality maps such as locality Rota-Baxter operators given by regularised sums and integrals. Using locality universal properties, we lift Rota-Baxter operators and branched sums to decorated rooted trees to build and study branched zeta functions associated with trees. These renormalised branched zeta functions are multiplicative on mutually independent trees.
85
132
1
10.4171/205-1/3
https://www.ems-ph.org/doi/10.4171/205-1/3
The scrambling operators applied to multizeta algebra and singular perturbation analysis
Jean
Ecalle
Université Paris-Sud, Université de Paris-Saclay, Orsay, France
General
The present paper addresses two seemingly unrelated topics – the analysis of singular-and-singularly-perturbed differential systems; and the arithmetics of multizetas – but with a strong unifying thread, provided by the three scrambling operators. The operators in question – scram, viscram, discram – properly belong to the field of combinatorics and mould algebra. Their properties are many, but one stands out: generating rich symmetries and sophisticated operations out of poorer or more elementary ones. The formal solutions of singular differential systems, when expanded in inverse-power series of the 'critical variable' $z$, tend to exhibit divergence, but of a regular and well-understood type: resummable and resurgent, with a resurgence regime completely governed by the now classical Bridge equation. When one introduces a singular perturbation parameter $\epsilon$ and expands the solution in powers of the same, divergence and resurgence still rule the show, but the picture becomes incomparably more complex: the resurgence calls for two new Bridge equations, not one; the familiar Stokes constants make way for the radically different tessellation coefficients; and it takes the operator scram to fully unravel the mechanisms responsible for this new level of complexity. The closely related operators viscram and discram, on their part, render distinguished services in multizeta algebra, especially for dissecting what is arguably the most pivotal case: the bicoloured multizetas. For one thing, they assist in proving the independence of the standard system of bicolour generators. But their real contribution lies elsewhere. The fact is that, due to the simultaneous play of weigths $s_i\in \mathbb N^\ast$ and colours $\epsilon_i \in \frac{1}{k}\mathbb Z/\mathbb Z$, there exist for any given (large) total weight $s$, a huge number of $k$-coloured multizetas. Yet there is a saving grace: the double symmetry (known as arithmetical dimorphy) which constrains these multizetas induces so strong a rigidity that the whole information can be recovered from relatively sparse boundary data (somewhat like with harmonic or analytic functions). The phenomenon is particularly striking in the case of bicolours $(k=2)$ and their three satellites: the 'lower satellite' sa, with all degrees set equal to $0$; the 'first upper satellite' sa$^\ast$, with all colours (simultaneously) set equal to $0$ or $\frac{1}{2}$; and the 'second upper satellite' sa$^{\ast\ast}$, similar in shape to the first, but completely different in origin. We show, with ample assistance from viscram and discram, how each of these three satellite systems not only morphs into the other two, but also leads to the complete system of bicolours – each conversion finding its expression in remarkably explicit formulae.
133
325
1
10.4171/205-1/4
https://www.ems-ph.org/doi/10.4171/205-1/4
Quasi-shuffle algebras and applications
Michael
Hoffman
United States Naval Academy, Annapolis, USA
Quasi-shuffle product, Hopf algebra, interpolated multiple zeta value
Associative rings and algebras
Number theory
Quasi-shuffle algebras have been a useful tool in studying multiple zeta values and related quantities, including multiple polylogarithms, finite multiple harmonic sums, and $q$-multiple zeta values. Here we show that two ideas previously considered only for multiple zeta values, the interpolated product of S. Yamamoto and the symmetric sum theorem, can be generalized to any quasi-shuffle algebra.
327
348
1
10.4171/205-1/5
https://www.ems-ph.org/doi/10.4171/205-1/5
Planar binary trees in scattering amplitudes
Carlos
Mafra
University of Southampton, UK
General
These notes are a written version of my talk given at the CARMA workshop in June 2017, with some additional material. I presented a few concepts that have recently been used in the computation of tree-level scattering amplitudes (mostly using pure spinor methods but not restricted to it) in a context that could be of interest to the combinatorics community. In particular, I focused on the appearance of planar binary trees in scattering amplitudes and presented some curious identities obeyed by related objects, some of which are known to be true only via explicit examples.
349
365
1
10.4171/205-1/6
https://www.ems-ph.org/doi/10.4171/205-1/6
A study on prefixes of $c_2$ invariants
Karen
Yeats
University of Waterloo, Canada
Combinatorics
Quantum theory
This paper begins by reviewing recent progress that has been made by taking a combinatorial perspective on the $c_2$ invariant, an arithmetic graph invariant with connections to Feynman integrals. Then it proceeds to report on some recent calculations of $c_2$ invariants for two families of circulant graphs at small primes. These calculations support the idea that all possible finite sequences appear as initial segments of $c_2$ invariants, in contrast to their apparent sparsity on small graphs.
367
383
1
10.4171/205-1/7
https://www.ems-ph.org/doi/10.4171/205-1/7
Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA)
Volume 1
Frédéric
Chapoton
Université de Strasbourg, France
Frédéric
Fauvet
Université de Strasbourg, France
Claudia
Malvenuto
Università di Roma La Sapienza, Italy
Jean-Yves
Thibon
Université Paris-Est Marne-la-Vallée, France
Quantum theory
Combinatorics
Ordinary differential equations
Dynamical systems and ergodic theory
05E, 81T15, 81T18, 81Q30, 34C20, 37C10, 18D50, 34M40, 34M60, 11M32, 30D60
Mathematics
Differential equations
Operad, Hopf algebra, algebraic combinatorics, moulds, renormalization, periods, multiple zeta values, resurgent functions, alien calculus, vector fields, diffeomorphisms
This is volume 1 of a 2-volume work comprising a total of 14 refereed research articles which stem from the CARMA Conference (Algebraic Combinatorics, Resurgence, Moulds and Applications), held at the Centre International de Rencontres Mathématiques in Luminy, France, from June 26 to 30, 2017. The conference did notably emphasise the role of Hopf algebraic techniques and related concepts (e.g. Rota–Baxter algebras, operads, Ecalle’s mould calculus) which have lately proved pervasive in combinatorics, but also in many other fields, from multiple zeta values to the algebraic study of control systems and the theory of rough paths. The volumes should be useful to researchers or graduate students in mathematics working in these domains and to theoretical physicists involved with resurgent functions and alien calculus.
2
28
2020
978-3-03719-204-7
978-3-03719-704-2
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/204
https://www.ems-ph.org/doi/10.4171/204
IRMA Lectures in Mathematics and Theoretical Physics
2523-5133
2523-5141
31
Shuffle quadri-algebras and concatenation
Mohamed
Belhaj Mohamed
Taibah University, Saudi Arabia, and Université de Sousse, Tunisia
Dominique
Manchon
Université Blaise Pascal, Aubière, France
Quadri-algebra, dendriform algebra, shuffle, concatenation, module-algebra
Combinatorics
Nonassociative rings and algebras
In this article, we study the shuffle quadri-algebra over some vector space. We prove the existence of some relations between the four quadri-algebra laws which constitute the shuffle product, the concatenation product and the deconcatenation coproduct. We also show that the shuffle quadri-algebra admits two module-algebra structures on itself endowed with the underlying associative algebra structure.
1
27
1
10.4171/204-1/1
https://www.ems-ph.org/doi/10.4171/204-1/1
Structure theorems for dendriform and tridendriform algebras
Emily
Burgunder
Université Paul Sabatier, Toulouse, France
Bérénice
Delcroix-Oger
Université Paris Diderot, Paris, France
Combinatorics
Nonassociative rings and algebras
We state new Cartier–Milnor–Moore Poincaré–Birkhoff–Witt theorems for dendriform and tridendriform structures. We introduce the terplicial coalgebra structure as an analogue in the tridendriform algebras of the duplicial co-structure for the dendriform case, and prove a rigidity theorem.
19
66
1
10.4171/204-1/2
https://www.ems-ph.org/doi/10.4171/204-1/2
A group-theoretical approach to conditionally free cumulants
Kurusch
Ebrahimi-Fard
Norwegian University of Science and Technology, Trondheim, Norway
Frédéric
Patras
Université Côte d'Azur, Nice, France
Free probability; moment-cumulant relations; c-free cumulants; combinatorial Hopf algebra; shuffle algebra; pre-Lie algebra
Associative rings and algebras
Functional analysis
In this work we extend the recently introduced group-theoretical approach to moment-cumulant relations in non-commutative probability theory to the notion of conditionally free cumulants. This approach is based on a particular combinatorial Hopf algebra which may be characterised as a non-cocommutative generalisation of the classical unshuffle Hopf algebra. Central to our work is the resulting non-commutative shuffle algebra structure on the graded dual. It implies an extension of the classical relation between the group of Hopf algebra characters and its Lie algebra of infinitesimal characters and, among others, the appearance of new forms of “adjoint actions” of the group on its Lie algebra which happens to play a key role in the new algebraic understanding of conditionally free cumulants.
67
92
1
10.4171/204-1/3
https://www.ems-ph.org/doi/10.4171/204-1/3
The Natural Growth Scale
Jean
Ecalle
Université Paris-Sud, Université de Paris-Saclay, Orsay, France
General
The present paper starts with the group of all germs of analytic self-mappings of $\mathbb R_{,+\infty}$ and concerns itself with its successive closures under (i) fractional iteration (ii) conjugation (iii) the solving of general composition equations. Rather than attempting a systematic treatment, we focus on the typical difficulties attendant upon these extensions. On the formal side, power series make way first for transseries, then for ultraseries, involving finite resp. transfinite iterates of the exponential. On the analysis side, the first casualties are convergence and analyticity: from the start, we have to face generic resurgence (multicritical but of a weakly polarising type) and, further down the road, generic cohesiveness (a natural and very inclusive extension of Denjoy quasi-analyticity). Nevertheless, none of these complications destroys the bi-constructive correspondence between the formal objects (series, transseries, ultraseries) and the geometric germs. We describe, and illustrate on numerous examples, the apparatus required for upholding this correspondence: mainly accelero-summation, which uses convolution-respecting integral transforms to ascend from one critical Borel plane to the next, and the so-called display, a semi-algebraic construct that supplements the genuine variable with a host of pseudo-variables and encapsulates in highly convenient form all the information about the resurgence pattern and Stokes constants of a given germ. We also devote three sections to the (non-linear) iso-differential operators which, on top of their surprising algebraic properties, are uniquely adapted to germ composition, the analysis of deep convexity, and the description of the universal asymptotics of very slow- or fast-growing germs. Lastly, we reflect on the seemingly unsurmountable indeterminacy inherent in the choice of transfinite exponential iterates, and on the implications of that indeterminacy for the natural growth scale(by which we mean, roughly speaking, the ultimate extension of our groups of non-oscillating germs): far from being the quintessential continuum that one would expect, the natural growth scale – on the formal as on the analysis side, in the large as well as locally – displays a granular, almost fractal-like structure.
93
223
1
10.4171/204-1/4
https://www.ems-ph.org/doi/10.4171/204-1/4
Realizations of Hopf algebras of graphs by alphabets
Loïc
Foissy
Université du Littoral Côte d’Opale, Calais, France
Combinatorial Hopf algebras; Feynman graphs; posets
Associative rings and algebras
Combinatorics
Order, lattices, ordered algebraic structures
We here give polynomial realizations of various Hopf algebras or bialgebras on Feynman graphs, graphs, posets or quasi-posets, that it to say injections of these objects into polynomial algebras generated by an alphabet. The alphabet here considered are totally quasi-ordered. The coproducts are given by doubling the alphabets; a second coproduct is defined by squaring the alphabets, and we obtain cointeracting bialgebras in the commutative case.
225
261
1
10.4171/204-1/5
https://www.ems-ph.org/doi/10.4171/204-1/5
Duplicial algebras, parking functions, and Lagrange inversion
Jean-Christophe
Novelli
Université de Marne-la-Vallée, France
Jean-Yves
Thibon
Université de Marne-la-Vallée, France
Operads, noncommutative symmetric functions, parking functions, Lagrange inversion
Category theory; homological algebra
Combinatorics
Order, lattices, ordered algebraic structures
Associative rings and algebras
We provide operadic interpretations for two Hopf subalgebras of the algebra of parking functions. The Catalan subalgebra is identified with the free duplicial algebra on one generator, and the Schröder subalgebra is interpreted by means of a new operad, which we call triduplicial. The noncommutative Lagrange inversion formula is then interpreted in terms of duplicial structures. The generic solution of the noncommutative inversion problem appears as the formal sum of all parking functions. This suggests that combinatorial generating functions derived by functional inversion should be obtainable by evaluating a suitable character on this generic solution. This idea is illustrated by means of the Narayana polynomials, of which we obtain bivariate “super-analogues” by lifting to parking functions a classical character of the algebra of symmetric functions. Other characters, such as evaluation of symmetric functions on a binomial element, are also discussed.
263
290
1
10.4171/204-1/6
https://www.ems-ph.org/doi/10.4171/204-1/6
The triduplicial operad is Koszul
Anthony
Mansuy
Université de Reims Champagne-Ardennes, Reims, France
General
291
297
1
10.4171/204-1/7
https://www.ems-ph.org/doi/10.4171/204-1/7
The Hopf algebra of integer binary relations
Vincent
Pilaud
École Polytechnique, Palaiseau, France
Viviane
Pons
Université Paris-Sud, Orsay, France
General
We construct a Hopf algebra on integer binary relations that contains under the same roof several well-known Hopf algebras related to the permutahedra and the associahedra: the Malvenuto–Reutenauer algebra on permutations, the Loday–Ronco algebra on planar binary trees, and the Chapoton algebras on ordered partitions and on Schröder trees. We also derive from our construction new Hopf structures on intervals of the weak order on permutations and of the Tamari order on binary trees.
299
344
1
10.4171/204-1/8
https://www.ems-ph.org/doi/10.4171/204-1/8
Handbook of Teichmüller Theory, Volume VII
Athanase
Papadopoulos
Université de Strasbourg, France
Functions of a complex variable
Algebraic geometry
Several complex variables and analytic spaces
Dynamical systems and ergodic theory
30F60, 32G15, 30C20, 14H60, 30C35, 30C62, 30C70, 30C75, 37F30, 57M50, 01A60, 01A55, 20F65, 20F67, 22E40, 30D30, 30D35, 30F45, 37F30, 53A30, 57M50
Functional analysis
Riemann surface, Teichmüller space, Deligne–Mumford compactification, universal Teichmüller space, complex geodesic, holomorphic differential, quadratic differential, projective structure, Mostow rigidity, hyperbolic structure, Fuchsian group, quasi-Fuchsian group, Kleinian group, ending lamination, Higgs bundle, higher Teichmüller theory, Douady-Earle extension, quasisymmetric map, quadiconformal mapping, type problem, conformal invariant, extremal length, extremal domain, Tissot indicatrix, almost analytic function, measurable Riemann Mapping Theorem, value distribution, Modulsatz, reduced module, line complex, Speiser tree
The present volume of the Handbook of Teichmüller theory is divided into three parts. The first part contains surveys on various topics in Teichmüller theory, including the complex structure of Teichmüller space, the Deligne–Mumford compactification of the moduli space, holomorphic quadratic differentials, Kleinian groups, hyperbolic 3-manifolds and the ending lamination theorem, the universal Teichmüller space, barycentric extensions of maps of the circle, and the theory of Higgs bundles. The second part consists of three historico-geometrical articles on Tissot (a precursor of the theory of quasiconfomal mappings), Grötzsch and Lavrentieff, the two main founders of the modern theory of quasiconformal mappings. The third part comprises English translations of five papers by Grötzsch, a paper by Lavrentieff, and three papers by Teichmüller. These nine papers are foundational essays on the theories of conformal invariants and quasiconformal mappings, with applications to conformal geometry, to the type problem and to Nevanlinna's theory. The papers are followed by commentaries that highlight the relations between them and between later works on the subject. These papers are not only historical documents; they constitute an invaluable source of ideas for current research in Teichmüller theory.
2
15
2020
978-3-03719-203-0
978-3-03719-703-5
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/203
https://www.ems-ph.org/doi/10.4171/203
IRMA Lectures in Mathematics and Theoretical Physics
2523-5133
2523-5141
30
Introduction to Teichmüller theory, old and new, VII
Athanase
Papadopoulos
Université de Strasbourg et CNRS, Strasbourg, France
Functions of a complex variable
1
17
1
10.4171/203-1/1
https://www.ems-ph.org/doi/10.4171/203-1/1
The Deligne–Mumford compactification and crystallographic groups
Yukio
Matsumoto
Gakushuin University, Tokyo, Japan
Functions of a complex variable
21
62
1
10.4171/203-1/2
https://www.ems-ph.org/doi/10.4171/203-1/2
Complex geometry of Teichmüller domains
Subhojoy
Gupta
Indian Institute of Science, Bangalore, India
Harish
Seshadri
Indian Institute of Science, Bangalore, India
Functions of a complex variable
63
87
1
10.4171/203-1/3
https://www.ems-ph.org/doi/10.4171/203-1/3
Holomorphic quadratic differentials in Teichmüller theory
Subhojoy
Gupta
Indian Institute of Science, Bangalore, India
Functions of a complex variable
89
124
1
10.4171/203-1/4
https://www.ems-ph.org/doi/10.4171/203-1/4
Mostow strong rigidity of locally symmetric spaces revisited
Lizhen
Ji
University of Michigan, Ann Arbor, USA
Functions of a complex variable
125
164
1
10.4171/203-1/5
https://www.ems-ph.org/doi/10.4171/203-1/5
Models of ends of hyperbolic 3-manifolds. A survey
Mahan
Mj
Tata Institute of Fundamental Research, Mumbai, India
Functions of a complex variable
165
193
1
10.4171/203-1/6
https://www.ems-ph.org/doi/10.4171/203-1/6
Universal Teichmüller space as a non-trivial example of infinite-dimensional complex manifolds
Armen
Sergeev
Russian Academy of Sciences, Moscow, Russian Federation
Functions of a complex variable
195
213
1
10.4171/203-1/7
https://www.ems-ph.org/doi/10.4171/203-1/7
Generalized conformal barycentric extensions of circle maps
Jun
Hu
Brooklyn College of CUNY, Brooklyn, USA
Functions of a complex variable
215
238
1
10.4171/203-1/8
https://www.ems-ph.org/doi/10.4171/203-1/8
Higgs bundles and higher Teichmüller spaces
Óscar
García Prada
Consejo Superior de Investigaciones Científicas, Madrid, Spain
Functions of a complex variable
239
285
1
10.4171/203-1/9
https://www.ems-ph.org/doi/10.4171/203-1/9
A note on Nicolas-Auguste Tissot: at the origin of quasiconformal mappings
Athanase
Papadopoulos
Université de Strasbourg et CNRS, Strasbourg, France
Functions of a complex variable
289
299
1
10.4171/203-1/10
https://www.ems-ph.org/doi/10.4171/203-1/10
Memories of Herbert Grötzsch
Reiner
Kühnau
Martin-Luther-Universität Halle-Wittenberg, Germany
Functions of a complex variable
301
315
1
10.4171/203-1/11
https://www.ems-ph.org/doi/10.4171/203-1/11
A note about Mikhaïl Lavrentieff and his world of analysis in the Soviet Union
Athanase
Papadopoulos
Université de Strasbourg et CNRS, Strasbourg, France
Galina
Sinkevich
Saint Petersburg State University of Architecture and Civil Engineering, Russian Federation
Functions of a complex variable
317
347
1
10.4171/203-1/12
https://www.ems-ph.org/doi/10.4171/203-1/12
A letter
Oswald
Teichmüller
Berlin, Germany
History and biography
351
353
1
10.4171/203-1/13
https://www.ems-ph.org/doi/10.4171/203-1/13
On some extremal problems of the conformal mapping
Herbert
Grötzsch
Leipzig, Germany
Functions of a complex variable
355
363
1
10.4171/203-1/14
https://www.ems-ph.org/doi/10.4171/203-1/14
On some extremal problems of the conformal mapping II
Herbert
Grötzsch
Leipzig, Germany
Functions of a complex variable
365
370
1
10.4171/203-1/15
https://www.ems-ph.org/doi/10.4171/203-1/15
On the distortion of schlicht non-conformal mappings and on a related extension of Picard’s theorem
Herbert
Grötzsch
Leipzig, Germany
Functions of a complex variable
371
374
1
10.4171/203-1/16
https://www.ems-ph.org/doi/10.4171/203-1/16
On the distortion of non-conformal schlicht mappings of multiply-connected schlicht regions
Herbert
Grötzsch
Leipzig, Germany
Functions of a complex variable
375
385
1
10.4171/203-1/17
https://www.ems-ph.org/doi/10.4171/203-1/17
On closest-to-conformal mappings
Herbert
Grötzsch
Leipzig, Germany
Functions of a complex variable
387
392
1
10.4171/203-1/18
https://www.ems-ph.org/doi/10.4171/203-1/18
On five papers by Herbert Grötzsch
Vincent
Alberge
Fordham University, Bronx, USA
Athanase
Papadopoulos
Université de Strasbourg et CNRS, Strasbourg, France
Functions of a complex variable
393
415
1
10.4171/203-1/19
https://www.ems-ph.org/doi/10.4171/203-1/19
On a class of continuous representations
Mikhaïl
Lavrentieff
Novosibirsk, Russian Federation
Functions of a complex variable
417
439
1
10.4171/203-1/20
https://www.ems-ph.org/doi/10.4171/203-1/20
A commentary on Lavrentieff’s paper "Sur une classe de représentations continues"
Vincent
Alberge
Fordham University, Bronx, USA
Athanase
Papadopoulos
Université de Strasbourg et CNRS, Strasbourg, France
Functions of a complex variable
441
451
1
10.4171/203-1/21
https://www.ems-ph.org/doi/10.4171/203-1/21
An application of quasiconformal mappings to the type problem
Oswald
Teichmüller
Berlin, Germany
Functions of a complex variable
453
461
1
10.4171/203-1/22
https://www.ems-ph.org/doi/10.4171/203-1/22
Investigations on conformal and quasiconformal mappings
Oswald
Teichmüller
Berlin, Germany
Functions of a complex variable
463
529
1
10.4171/203-1/23
https://www.ems-ph.org/doi/10.4171/203-1/23
Simple examples for value distribution
Oswald
Teichmüller
Berlin, Germany
Functions of a complex variable
531
542
1
10.4171/203-1/24
https://www.ems-ph.org/doi/10.4171/203-1/24
Teichmüller’s work on the type problem
Vincent
Alberge
Fordham University, Bronx, USA
Melkana
Brakalova-Trevithick
Fordham University, New York, USA
Athanase
Papadopoulos
Université de Strasbourg et CNRS, Strasbourg, France
Functions of a complex variable
543
560
1
10.4171/203-1/25
https://www.ems-ph.org/doi/10.4171/203-1/25
A Commentary on Teichmüller’s paper "Untersuchungen über konforme und quasikonforme Abbildungen"
Vincent
Alberge
Fordham University, Bronx, USA
Melkana
Brakalova-Trevithick
Fordham University, New York, USA
Athanase
Papadopoulos
Université de Strasbourg et CNRS, Strasbourg, France
Functions of a complex variable
561
583
1
10.4171/203-1/26
https://www.ems-ph.org/doi/10.4171/203-1/26
Value distribution theory and Teichmüller’s paper "Einfache Beispiele zur Wertverteilungslehre"
Athanase
Papadopoulos
Université de Strasbourg et CNRS, Strasbourg, France
Functions of a complex variable
585
603
1
10.4171/203-1/27
https://www.ems-ph.org/doi/10.4171/203-1/27
Hyperbolic Flows
Todd
Fisher
Brigham Young University, Provo, USA
Boris
Hasselblatt
Tufts University, Medford, USA
Dynamical systems and ergodic theory
37D40, 37D20; 37A30, 37A35
Differential equations
hyperbolic, hyperbolicity, flow, ergodic theory, topological dynamics, rigidity, expansiveness, shadowing, specification, geodesic flow, Anosov flow, Axiom A, entropy, equilibrium states, stable manifold, topological pressure, symbolic flows, Markov partitions
The origins of dynamical systems trace back to flows and differential equations, and this is a modern text and reference on dynamical systems in which continuous-time dynamics is primary. It addresses needs unmet by modern books on dynamical systems, which largely focus on discrete time. Students have lacked a useful introduction to flows, and researchers have difficulty finding references to cite for core results in the theory of flows. Even when these are known substantial diligence and consultation with experts is often needed to find them. This book presents the theory of flows from the topological, smooth, and measurable points of view. The first part introduces the general topological and ergodic theory of flows, and the second part presents the core theory of hyperbolic flows as well as a range of recent developments. Therefore, the book can be used both as a textbook – for either courses or self-study – and as a reference for students and researchers. There are a number of new results in the book, and many more are hard to locate elsewhere, often having appeared only in the original research literature. This book makes them all easily accessible and does so in the context of a comprehensive and coherent presentation of the theory of hyperbolic flows.
12
10
2019
978-3-03719-200-9
978-3-03719-700-4
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/200
https://www.ems-ph.org/doi/10.4171/200
Zurich Lectures in Advanced Mathematics
Gösta Mittag-Leffler and Vito Volterra. 40 Years of Correspondence
Frédéric
Jaëck
Ecole Normale Supérieure, Paris, France
Laurent
Mazliak
Université Pierre et Marie Curie, Paris, France
Emma
Sallent Del Colombo
Universitat de Barcelona, Spain
Rossana
Tazzioli
Université Lille 1, Villeneuve-d'Ascq, France
History and biography
01-XX
Mathematics and science
Mittag-Leffler, Volterra, correspondence, mathematics, history of mathematics
The present book contains the voluminous correspondence exchanged between the Swedish mathematician Gösta Mittag-Leffler and his younger Italian colleague Vito Volterra spanning a period of almost forty years at the end of the 19th and beginning of the 20th centuries. The relationship between the two men is remarkable for both personal and scientific reasons. Mittag-Leffler met Volterra for the first time as a brilliant young student of Ulisse Dini in Pisa. He was soon captivated by the creativity and the skills of the young man, and eventually became his mentor. Being himself at the center of a major scientific network, Mittag-Leffler introduced Volterra to the major mathematicians of the time, especially the Germans (Weierstrass, Klein, Cantor…) and French (Darboux, Jordan…). In a few years, Volterra became the most prominent Italian mathematician and forged his own network of scientists all over Europe, and even in the United States which he was one of the first major European mathematicians to visit. Despite their difference in age, both men developed a deep and faithful friendship and their letters reflect the variety of themes of their exchanges. Of course, mathematics was the most prominent, and both men often used the letters as a first draft of their ideas and the addressee as a first judge of their soundness. Besides mathematics, they also touched upon many aspects of both private and public life: matrimony, children, holidays, politics and so on. This vast set of letters affords the reader a general overview of mathematical life at the turn of the 19th century and an appreciation of the European intellectual spirit which came to an end, or at least suffered a drastic turn, when the Great War broke out. Volterra and Mittag-Leffler’s exchanges illustrate how general analysis, especially functional analysis, gained a dramatic momentum during those years, and how Volterra became one of the major leaders of the topic, opening the path for several fundamental developments over the following decades. Through the letters one can follow the institutional career and scientific activity of both Volterra and Mittag-Leffler who shared many details about their situation. The four editors are all specialists in the history of mathematics of the considered period. An extensive general introduction to the correspondence explains the context and the conditions in which it was developed. Moreover, the original letters are annotated with a large number of footnotes, which provide a broader cultural picture from these captivating documents.
11
20
2019
978-3-03719-199-6
978-3-03719-699-1
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/199
https://www.ems-ph.org/doi/10.4171/199
Heritage of European Mathematics
2523-5214
2523-5222
$t$-Motives: Hodge Structures, Transcendence and Other Motivic Aspects
Gebhard
Böckle
Universität Heidelberg, Germany
David
Goss
The Ohio State University, Columbus, USA
Urs
Hartl
Universität Münster, Germany
Matthew
Papanikolas
Texas A&M University, College Station, USA
Number theory
Commutative rings and algebras
11G09; 11J93, 11R58, 13A35
Number theory
Drinfeld modules, $t$-motives, Anderson $t$-modules, transcendence, Hodge-Pink-structures
This volume contains research and survey articles on Drinfeld modules, Anderson $t$-modules and $t$-motives. Much material that had not been easily accessible in the literature is presented here, for example the cohomology theories and Pink’s theory of Hodge structures attached to Drinfeld modules and $t$-motives. Also included are survey articles on the function field analogue of Fontaine’s theory of $p$-adic crystalline Galois representations and on transcendence methods over function fields, encompassing the theories of Frobenius difference equations, automata theory, and Mahler’s method. In addition, this volume contains a small number of research articles on function field Iwasawa theory, 1-$t$-motifs, and multizeta values. This book is a useful source for learning important techniques and an effective reference for all researchers working in or interested in the area of function field arithmetic, from graduate students to established experts.
3
31
2020
978-3-03719-198-9
978-3-03719-698-4
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/198
https://www.ems-ph.org/doi/10.4171/198
EMS Series of Congress Reports
2523-515X
2523-5168
A rapid introduction to Drinfeld modules, $t$-modules, and $t$-motives
W. Dale
Brownawell
The Pennsylvania State University, University Park, USA
Matthew
Papanikolas
Texas A&M University, College Station, USA
Number theory
3
30
1
10.4171/198-1/1
https://www.ems-ph.org/doi/10.4171/198-1/1
Pink’s theory of Hodge structures and the Hodge conjecture over function fields
Urs
Hartl
Westfälische Wilhelms-Universität Münster, Germany
Ann-Kristin
Juschka
Universität Heidelberg, Germany
Number theory
Commutative rings and algebras
Algebraic geometry
In 1997 Richard Pink has clarified the concept of Hodge structures over function fields in positive characteristic, which today are called Hodge-Pink structures. They form a neutral Tannakian category over the underlying function field. He has defined Hodge realization functors from the uniformizable abelian $t$-modules and $t$-motives of Greg Anderson to Hodge-Pink structures. This allows one to associate with each uniformizable $t$-motive a Hodge-Pink group, analogous to the Mumford-Tate group of a smooth projective variety over the complex numbers. It further enabled Pink to prove the analog of the Mumford-Tate Conjecture for Drinfeld modules. Moreover, based on unpublished work of Pink and the first author, the second author proved in her Diploma thesis that the Hodge-Pink group equals the motivic Galois group of the $t$-motive as defined by Papanikolas and Taelman. This yields a precise analog of the famous Hodge Conjecture, which is an outstanding open problem for varieties over the complex numbers. In this report we explain Pink's results on Hodge structures and the proof of the function field analog of the Hodge conjecture. The theory of $t$-motives has a variant in the theory of dual $t$-motives. We clarify the relation between $t$-motives, dual $t$-motives and $t$-modules. We also construct cohomology realizations of abelian $t$-modules and (dual) $t$-motives and comparison isomorphisms between them generalizing Gekeler's de Rham isomorphism for Drinfeld modules.
31
182
1
10.4171/198-1/2
https://www.ems-ph.org/doi/10.4171/198-1/2
Local shtukas, Hodge–Pink structures and Galois representations
Urs
Hartl
Westfälische Wilhelms-Universität Münster, Germany
Wansu
Kim
Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Republic of Korea
Number theory
We review the analog of Fontaine's theory of crystalline $p$-adic Galois representations and their classification by weakly admissible filtered isocrystals in the arithmetic of function fields over a finite field. There crystalline Galois representations are replaced by the Tate modules of so-called local shtukas. We prove that the Tate module functor is fully faithful. In addition to this étale realization of a local shtuka we discuss also the de Rham and the crystalline cohomology realizations and construct comparison isomorphisms between these realizations. We explain how local shtukas and these cohomology realizations arise from Drinfeld modules and Anderson's $t$-motives. As an application we construct equi-characteristic crystalline deformation rings, establish their rigid-analytic smoothness and compute their dimension.
183
259
1
10.4171/198-1/3
https://www.ems-ph.org/doi/10.4171/198-1/3
Frobenius difference equations and difference Galois groups
Chieh-Yu
Chang
National Tsing Hua University, Hsinchu, Taiwan
Difference equations, difference Galois groups, Drinfeld modules, periods, $\zeta$-values, $\G$-values, $t$-motives, transcendence
Number theory
261
295
1
10.4171/198-1/4
https://www.ems-ph.org/doi/10.4171/198-1/4
An introduction to Mahler’s method for transcendence and algebraic independence
Federico
Pellarin
Université Jean-Monnet, Saint-Étienne, France
Number theory
297
349
1
10.4171/198-1/5
https://www.ems-ph.org/doi/10.4171/198-1/5
Automata methods in transcendence
Dinesh
Thakur
University of Rochester, USA
Automata, periods, Drinfeld modules, special values
Number theory
Computer science
The purpose of this expository article is to explain diverse new tools that automata theory provides to tackle transcendence problems in function field arithmetic. We collect and explain various useful results scattered in computer science, formal languages, logic literature and explain how they can be fruitfully used in number theory, dealing with transcendence, refined transcendence and classification problems.
351
372
1
10.4171/198-1/6
https://www.ems-ph.org/doi/10.4171/198-1/6
Aspects of Iwasawa theory over function fields
Andrea
Bandini
Università degli Studi di Pisa, Italy
Francesc
Bars
Universitat Autònoma de Barcelona, Bellaterra (Barcelona), Spain
Ignazio
Longhi
National Taiwan University, Taipei, Taiwan
Iwasawa Main Conjecture, global function fields, $L$-functions, Selmer groups, class groups, Bernoulli–Carlitz numbers
Number theory
Algebraic geometry
We consider $\mathbb Z_p^{\mathbb{N}}$-extensions $\mathcal F$ of a global function field $F$ and study various aspects of Iwasawa theory with emphasis on the two main themes already (and still) developed in the number fields case as well. When dealing with the Selmer group of an abelian variety $A$ defined over $F$, we provide all the ingredients to formulate an Iwasawa Main Conjecture relating the Fitting ideal and the $p$-adic $L$-function associated to $A$ and $\mathcal F$. We do the same, with characteristic ideals and $p$-adic $L$-functions, in the case of class groups (using known results on characteristic ideals and Stickelberger elements for $\mathbb Z_p^d$-extensions). The final section provides more details for the cyclotomic $\mathbb Z_p^{\mathbb{N}}$-extension arising from the torsion of the Carlitz module: in particular, we relate cyclotomic units with Bernoulli–Carlitz numbers by a Coates–Wiles homomorphism.
375
416
1
10.4171/198-1/7
https://www.ems-ph.org/doi/10.4171/198-1/7
1-$t$-Motifs
Lenny
Taelman
University of Amsterdam, Netherlands
Number theory
We show that the module of rational points on an abelian $t$\dash module $E$ is canonically isomorphic with the module $\mathrm {Ext}^1(M_E, K[t])$ of extensions of the trivial $t$-motif $K[t] $ by the $t$-motif $M_E$ associated with $E$. This generalizes prior results of Anderson and Thakur, Papanikolas and Ramachandran, and Woo. In case $E$ is uniformizable we show that this extension module is canonically isomorphic with the corresponding extension module of Pink–Hodge structures. This situation is formally very similar to Deligne's theory of 1-motifs and we have tried to build up the theory in a way that makes this analogy as clear as possible.
417
439
1
10.4171/198-1/8
https://www.ems-ph.org/doi/10.4171/198-1/8
Multizeta in function field arithmetic
Dinesh
Thakur
University of Rochester, USA
Mixed motives, extensions, Drinfeld modules, shuffle
Number theory
This is a brief report on recent work of the author (some joint with Greg Anderson) and his student on multizeta values for function fields. This includes definitions, proofs and conjectures on the relations, period interpretation in terms of mixed Carlitz–Tate $t$-motives and related motivic aspects. We also verify Taelman's recent conjectures in special cases.
441
452
1
10.4171/198-1/9
https://www.ems-ph.org/doi/10.4171/198-1/9
Spectral Structures and Topological Methods in Mathematics
Michael
Baake
Universität Bielefeld, Germany
Friedrich
Götze
Universität Bielefeld, Germany
Werner
Hoffmann
Universität Bielefeld, Germany
Global analysis, analysis on manifolds
Number theory
Group theory and generalizations
Convex and discrete geometry
Primary 58J65, 52C23, 20F65, 11M41; secondary 46L54, 60J45, 35C07, 35Q55, 43A25, 20F36, 11R42, 14L05
Mathematics and science
Calculus + mathematical analysis
Universal distributions, free probability, Markov processes, Schrödinger operators, heat kernel, spatial ecology, metastability, numerical analysis, critical regularity, aperiodic order, dynamical systems, special Kähler structure, non-crossing partitions, localising subcategory, braided groups, zeta functions, subgroup growth, representation growth, Brumer–Stark conjecture, p-divisible groups
This book is a collection of survey articles about spectral structures and the application of topological methods bridging different mathematical disciplines, from pure to applied. The topics are based on work done in the Collaborative Research Centre (SFB) 701. Notable examples are non-crossing partitions, which connect representation theory, braid groups, non-commutative probability as well as spectral distributions of random matrices. The local distributions of such spectra are universal, also representing the local distribution of zeros of $L$-functions in number theory. An overarching method is the use of zeta functions in the asymptotic counting of sublattices, group representations etc. Further examples connecting probability, analysis, dynamical systems and geometry are generating operators of deterministic or stochastic processes, stochastic differential equations, and fractals, relating them to the local geometry of such spaces and the convergence to stable and semi-stable states.
7
20
2019
978-3-03719-197-2
978-3-03719-697-7
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/197
https://www.ems-ph.org/doi/10.4171/197
EMS Series of Congress Reports
2523-515X
2523-5168
Convergence and asymptotic approximations to universal distributions in probability
Friedrich
Götze
Universität Bielefeld, Germany
Holger
Kösters
Universität Rostock, Germany
Global analysis, analysis on manifolds
The limiting distributions of functionals depending on a large number of independent random variables of comparable size are often universal, leading to a vast number of convergence and approximation results. We discuss some general principles that have emerged in recent years. Examples include classical and entropic central limit theorems in classical and free probability, distributions of zeros of random polynomials of high degree and related distributions of algebraic numbers, as well as global and local universality results for spectral distributions of random matrices.
1
28
1
10.4171/197-1/1
https://www.ems-ph.org/doi/10.4171/197-1/1
Kolmogorov operators and SPDEs
Michael
Röckner
Universität Bielefeld, Germany
Global analysis, analysis on manifolds
The purpose of this paper is to survey a number of selected results on Kolmogorov operators and SPDEs. It consists of two parts: One part is related to Kolmogorov operators and is devoted to the corresponding linear Fokker–Planck–Kolmogorov equations (see [11]), the other part is about three key results about SPDEs obtained resp. published during the last funding period, namely an existence and uniqueness result for $L^2$-initial data for the stochastic total variation flow, a new approach to SPDEs and a pathwise uniqueness result of SDEs on Hilbert spaces with a merely bounded drift part.
29
53
1
10.4171/197-1/2
https://www.ems-ph.org/doi/10.4171/197-1/2
Analysis and stochastic processes on metric measure spaces
Alexander
Grigor'yan
Universität Bielefeld, Germany
Global analysis, analysis on manifolds
This contribution deals with the properties of certain differential and nonlocal operators on various spaces, with the emphasis on the relationship between the analytic properties of the operators in question and the geometric properties of the underlying space. In most situations, these operators are Markov generators. In such cases, we are also concerned with probabilistic aspects, such as the path properties of the corresponding Markov process.
55
73
1
10.4171/197-1/3
https://www.ems-ph.org/doi/10.4171/197-1/3
Markov evolutions in spatial ecology: From microscopic dynamics to kinetics
Yuri
Kondratiev
Universität Bielefeld, Germany
Oleksandr
Kutoviy
Universität Bielefeld, Germany
Pavlo
Tkachov
Universität Bielefeld, Germany
Global analysis, analysis on manifolds
In this summary, we construct Markov statistical dynamics for a class of birth-and-death ecological models in the continuum. Mesoscopic scaling limits for these dynamics lead to the kinetic equations for the density of a population. The resulting evolution equations are non-local and non-linear ones. We study properties of solutions to kinetic equations which strongly depend on characteristics of the models considered.
75
105
1
10.4171/197-1/4
https://www.ems-ph.org/doi/10.4171/197-1/4
Metastability in randomly perturbed dynamical systems: Beyond large-deviation theory
Barbara
Gentz
Universität Bielefeld, Germany
Global analysis, analysis on manifolds
We address the question of noise-induced transitions in continuous-time dynamical systems with special emphasis on aspects which go beyond standard large-deviation theory. After reviewing the classical Wentzell–Freidlin theory, we discuss the subexponential asymptotics of transition times between potential wells, transitions between stationary states in parabolic stochastic partial differential equations (SPDEs), first-exit from a domain with characteristic boundary, and the effect of noise on so-called mixed-mode oscillations.
107
128
1
10.4171/197-1/5
https://www.ems-ph.org/doi/10.4171/197-1/5
Computation and stability of waves in equivariant evolution equations
Wolf-Jürgen
Beyn
Universität Bielefeld, Germany
Denny
Otten
Universität Bielefeld, Germany
Global analysis, analysis on manifolds
Travelling and rotating waves are ubiquitous phenomena observed in time dependent PDEs that model the combined effect of dissipation and nonlinear interaction. From an abstract viewpoint, they appear as relative equilibria of an equivariant evolution equation. In numerical computations, the freezing method takes advantage of this structure by splitting the evolution of the PDE into the dynamics on the underlying Lie group and on some reduced phase space. The approach raises a series of questions which were answered to a certain extent: linear stability implies nonlinear (asymptotic) stability, persistence of stability under discretisation, analysis and computation of spectral structures, first versus second order evolution systems, well-posedness of partial differential algebraic equations, spatial decay of wave profiles and truncation to bounded domains, analytical and numerical treatment of wave interactions, relation to connecting orbits in dynamical systems. A further numerical problem related to this topic will be discussed, namely the solution of nonlinear eigenvalue problems via a contour method.
129
158
1
10.4171/197-1/6
https://www.ems-ph.org/doi/10.4171/197-1/6
Initial value problems for nonlinear dispersive equations at critical regularity
Sebastian
Herr
Universität Bielefeld, Germany
Global analysis, analysis on manifolds
Global regularity results for nonlinear dispersive equations hinge on a thorough understanding of the Cauchy problem in spaces of functions of low regularity. This is most challenging in scale invariant regimes as solutions interact strongly on multiple frequency-scales. Here, some recent progress on the critical well-posedness theory will be reviewed, with a focus on nonlinear Schrödinger and Dirac equations.
159
182
1
10.4171/197-1/7
https://www.ems-ph.org/doi/10.4171/197-1/7
Variational solutions to nonlocal problems
Moritz
Kassmann
Universität Bielefeld, Germany
Global analysis, analysis on manifolds
We present recent results on nonlocal operators acting on real-valued functions that are defined on subsets of $\mathbb R^d$. The operators under consideration exhibit a fractional order of differentiability and have attracted a lot of attention in the last twenty years. It turns out that several ideas and approaches developed for the study of partial differential operators of second order can be applied after suitable modifications. In this report, we explain similarities and differences with regard to the well-established theory for differential operators of second order. We concentrate on stationary linear symmetric operators in variational form.
183
196
1
10.4171/197-1/8
https://www.ems-ph.org/doi/10.4171/197-1/8
Spectral and arithmetic structures in aperiodic order
Michael
Baake
Universität Bielefeld, Germany
Franz
Gähler
Universität Bielefeld, Germany
Christian
Huck
Universität Bielefeld, Germany
Peter
Zeiner
Xiamen University Malaysia, Sepang, Selangor, Malaysia
Global analysis, analysis on manifolds
Systems with aperiodic order can display a variety of arithmetic, combinatorial and spectral phenomena, some of which are reviewed and discussed here. At the same time, the underlying compact tiling spaces can be compared via their topological and spectral invariants. The latter are explicitly computable for substitution systems and provide an important tool for their classification.
197
220
1
10.4171/197-1/9
https://www.ems-ph.org/doi/10.4171/197-1/9
Affine special Kähler structures in real dimension two
Martin
Callies
Universität Bielefeld, Germany
Andriy
Haydys
Universität Freiburg, Germany
Global analysis, analysis on manifolds
We review properties of affine special Kähler structures focusing on singularities of such structures in the simplest case of real dimension two. We describe all possible isolated singularities and corresponding monodromies of the flat symplectic connection, which is a part of a special Kähler structure, near a singularity. Beside numerous local examples, we construct continuous families of special Kähler structures with isolated singularities on the projective line.
221
233
1
10.4171/197-1/10
https://www.ems-ph.org/doi/10.4171/197-1/10
Non-crossing partitions
Barbara
Baumeister
Universität Bielefeld, Germany
Kai-Uwe
Bux
Universität Bielefeld, Germany
Friedrich
Götze
Universität Bielefeld, Germany
Dawid
Kielak
Universität Bielefeld, Germany
Henning
Krause
Universität Bielefeld, Germany
Global analysis, analysis on manifolds
Non-crossing partitions have been a staple in combinatorics for quite some time. More recently, they have surfaced (sometimes unexpectedly) in various other contexts from free probability to classifying spaces of braid groups. Also, analogues of the non-crossing partition lattice have been introduced. Here, the classical noncrossing partitions are associated to Coxeter and Artin groups of type A$_n$, which explains the tight connection to the symmetric groups and braid groups. We shall outline those developments.
235
274
1
10.4171/197-1/11
https://www.ems-ph.org/doi/10.4171/197-1/11
The derived category of the projective line
Henning
Krause
Universität Bielefeld, Germany
Greg
Stevenson
University of Glasgow, UK
Global analysis, analysis on manifolds
In this chapter, we discuss the localising subcategories of the derived category of quasi-coherent sheaves on the projective line over a field. We provide a complete classification of all such subcategories which arise as the kernel of a cohomological functor to a Grothendieck category.
275
297
1
10.4171/197-1/12
https://www.ems-ph.org/doi/10.4171/197-1/12
Higher finiteness properties of braided groups
Kai-Uwe
Bux
Universität Bielefeld, Germany
Global analysis, analysis on manifolds
The properties of a group to be finitely generated or finitely presentable are the first two instances in a sequence of so called higher finiteness properties defined in terms of skeletons of classifying spaces. The study of higher finiteness properties is a prime example of how one can use a nice action of a group on a topological space to better understand the group. We shall illustrate this method in detail using the braided Thompson group $V^{\mathrm {br}}$ as an example.
299
321
1
10.4171/197-1/13
https://www.ems-ph.org/doi/10.4171/197-1/13
Zeta functions and the trace formula
Werner
Hoffmann
Universität Bielefeld, Germany
Global analysis, analysis on manifolds
Prehomogeneous vector spaces provide a framework for the method of analytic continuation of zeta integrals due to Hecke and Tate. We will describe instances where convergence can only be achieved by truncation. Special values of such zeta integrals appear in the Arthur–Selberg trace formula, and their study is relevant in connection with recent ideas in the Langlands program.
323
343
1
10.4171/197-1/14
https://www.ems-ph.org/doi/10.4171/197-1/14
Zeta functions of groups and rings – functional equations and analytic uniformity
Christopher
Voll
Universität Bielefeld, Germany
Global analysis, analysis on manifolds
Zeta functions are widely used tools in the study of asymptotic properties of infinite groups and rings, in particular their subobject and representation growth. We survey recent results on arithmetic and asymptotic features of such functions, focussing on various classes of subobject zeta functions, in particular submodule zeta functions associated with nilpotent algebras of endomorphisms, and representation zeta functions associated to arithmetic groups, specifically finitely generated nilpotent groups.
345
363
1
10.4171/197-1/15
https://www.ems-ph.org/doi/10.4171/197-1/15
Conjectures of Brumer, Gross and Stark
Andreas
Nickel
Universität Duisburg-Essen, Germany
Global analysis, analysis on manifolds
This chapter gives an introduction to generalisations of conjectures of Brumer and Stark on the annihilator of the class group of a number field. We review the relation to the equivariant Tamagawa number conjecture, the main conjecture of Iwasawa theory for totally real fields, and a conjecture of Gross on the behaviour of $p$-adic Artin $L$-functions at zero.
365
388
1
10.4171/197-1/16
https://www.ems-ph.org/doi/10.4171/197-1/16
Displays and $p$-divisible groups
Thomas
Zink
Universität Bielefeld, Germany
Global analysis, analysis on manifolds
The theory of displays is a Dieudonné theory for formal $p$-divisible groups which is an equivalence of categories over an arbitrary $p$-adic ring. Over a more restricted class of rings one obtains a classification of all $p$-divisible groups. We explain basic ideas and some recent results of this theory. The last paragraph ameliorates the discussion of isogenies of displays found in the literature.
389
408
1
10.4171/197-1/17
https://www.ems-ph.org/doi/10.4171/197-1/17
Eighteen Essays in Non-Euclidean Geometry
Vincent
Alberge
Fordham University, Bronx, USA
Athanase
Papadopoulos
Université de Strasbourg, France
Mathematical logic and foundations
Geometry
Convex and discrete geometry
Differential geometry
03B30, 51A05, 51F15, 51F20, 51M04, 51M05, 51M09, 51M10, 51M16, 51M20, 51M25, 51N15, 52A15, 52A55, 52B15, 52C25, 05C62, 53-01, 53A05, 53A30, 53A35, 53C35, 53C45
Mathematics
Mathematical logic
Non-Euclidean geometry, spherical geometry, hyperbolic geometry, Busemann type geometry, curvature, geographical map, non-euclidean area, non-euclidean volume, Brahmagupta’s formula, Ptolemy’s theorem, Casey’s theorem, Sforza’s formula, Seidel’s problem, infinitesimal rigidity, static rigidity, Pogorelov map, Maxwell–Cremona correspondence, exterior hyperbolic geometry, de Sitter geometry, non-Euclidean conics, bifocal properties, focus-directrix properties, pencils of conics, projective geometry, convexity, duality, transition, Hermitian trigonometry, complex projective trigonometry, shape invariant, metric plane projective-metric plane
This book consists of a series of self-contained essays in non-Euclidean geometry in a broad sense, including the classical geometries of constant curvature (spherical and hyperbolic), de Sitter, anti-de Sitter, co-Euclidean, co-Minkowski, Hermitian geometries, and some axiomatically defined geometries. Some of these essays deal with very classical questions and others address problems that are at the heart of present day research, but all of them are concerned with fundamental topics. All the essays are self-contained and most of them can be understood by the general educated mathematician. They should be useful to researchers and to students of non-Euclidean geometry, and they are intended to be references for the various topics they present.
3
31
2019
978-3-03719-196-5
978-3-03719-696-0
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/196
https://www.ems-ph.org/doi/10.4171/196
IRMA Lectures in Mathematics and Theoretical Physics
2523-5133
2523-5141
29
Area in non-Euclidean geometry
Norbert
A’Campo
Universität Basel, Switzerland
Athanase
Papadopoulos
Université de Strasbourg, France
Geometry
We start by recalling the proof of the classical theorem of Girard on the area of a spherical triangle in terms of its angle sum, and its analogue in hyperbolic geometry. We then use a formula of Euler for the area of a spherical triangle in terms of side lengths and its analogue in hyperbolic geometry in order to give an equality for the distance between the midpoints of two sides of a spherical (respectively hyperbolic) triangle, in terms of the third side. These equalities give quantitative versions of the positivity (respectively negativity) of the curvature in the sense of Busemann. At the same time, we present several other results related to area in non-Euclidean geometry together with historical comments. The scope of the ideas that we survey in this short paper, ranging from Menelaus of Alexandria (1st-2nd century A.D.) to William Thurston, mentioning in-between works of Albert Girard (17th c.), Euler, Lambert, Lagrange, Lobachevsky, Busemann and others, gives an idea of the richness of the notion of area in non-Euclidean geometry.
3
25
1
10.4171/196-1/1
https://www.ems-ph.org/doi/10.4171/196-1/1
The area formula for hyperbolic triangles
Elena
Frenkel
Université de Strasbourg, France
Weixu
Su
Fudan University, Shanghai, China
Geometry
We prove an area formula in terms of side lengths for hyperbolic triangles. The proof is analogous to a proof given by Leonhard Euler in the spherical case. We take this opportunity to prove other results in hyperbolic geometry using the variational techniques that Euler introduced in his work on spherical geometry. We consider in particular the famous Lexell Problem, that is, the problem of finding the locus of vertices of triangles of fixed area and fixed basis.
27
46
1
10.4171/196-1/2
https://www.ems-ph.org/doi/10.4171/196-1/2
On a problem of Schubert in hyperbolic geometry
Vincent
Alberge
Fordham University, Bronx, USA
Elena
Frenkel
Université de Strasbourg, France
Geometry
We solve in two different ways a problem in hyperbolic geometry, namely, to find for a given segment on a geodesic line and a given hypercycle, the point(s) on the hypercycle for which the area of the corresponding triangle is maximal or minimal.
47
56
1
10.4171/196-1/3
https://www.ems-ph.org/doi/10.4171/196-1/3
On a theorem of Lambert: medians in spherical and hyperbolic geometries
Himalaya
Senapati
Chennai Mathematical Institute, Siruseri, India
Geometry
57
65
1
10.4171/196-1/4
https://www.ems-ph.org/doi/10.4171/196-1/4
Inscribing a triangle in a circle in spherical geometry
Himalaya
Senapati
Chennai Mathematical Institute, Siruseri, India
Geometry
We consider the geometrical problem of constructing a triangle using a straightedge and compass so that its three vertices lie on a given circle and its three sides when produced pass through three given points. This problem was formulated and studied by Pappus of Alexandria in his Collection in the special case where the three given points are aligned (Proposition 117 of Book VII of the Collection). Lagrange gave an algebraic solution to this problem (1776). Euler gave another solution and commented on the case where the given circle and the three given points are on a sphere (1780). In this chapter, after recalling the solution due to Lagrange, we study the same problem on the sphere.
67
79
1
10.4171/196-1/5
https://www.ems-ph.org/doi/10.4171/196-1/5
Monotonicity in spherical and hyperbolic triangles
Himalaya
Senapati
Chennai Mathematical Institute, Siruseri, India
Geometry
We start by recalling Proposition 27 of Menelaus’ Spherics, a treatise on spherical geometry by Menelaus of Alexandria (1st–2nd century CE). This is a comparison theorem between the base of a spherical triangle and the great circle arc joining the midpoints of the two legs. The theorem expresses a property which later became known as Busemann's criterion for positive curvature in metric spaces. Propositions 28 and 29 that follow in the same treatise are comparison theorems of a similar sort, but comparing angles instead of edges when a spherical triangle is divided by a great circle arc joining midpoints of two sides. In this comparison of angles, Menelaus assumed one of the angles of the triangle to be at least equal to a right angle. In this chapter, we prove a monotonicity result in the lineage of Propositions 28 and 29. More precisely, we prove that in a spherical triangle $ABC$ with $CB>CA$, if the sides $CA$ and $CB$ are increased while holding the angle $C$ and the ratio $CA/CB$ constant, the angle $\widehat{CAB}$ increases monotonically. As a corollary, we prove parts of Propositions 28 and 29 of the \emph{Spherics} without any restriction on angles: the angle subtended at the base of a spherical triangle is greater than its homologue subtended at the geodesic joining the midpoints of the legs if the adjacent leg is shorter than the farther one. We then extend this result to hyperbolic geometry where we prove that the angle $\widehat{CAB}$ decreases monotonically.
81
91
1
10.4171/196-1/6
https://www.ems-ph.org/doi/10.4171/196-1/6
De Tilly’s mechanical view on hyperbolic and spherical geometries
Dmitriy
Slutskiy
Université de Cergy-Pontoise, France
Geometry
In this chapter, we describe a kinematic approach developed by J.-M. de Tilly for the computation of the length of a curve at distance $r$ from a geodesic (function $\eq(r)$) and of the length of a circle of radius $r$ (function ${\circ}(r)$) in the $2$-plane of any constant curvature $K$, $K\in\mathbb{R}$. We study the rotation and the translation of a segment and of a triangle to obtain various formulae relating the functions $\eq(r)$ and ${\circ}(r)$. As a corollary we give an elementary proof of the Laws of Sines and Cosines in hyperbolic and spherical spaces.
93
111
1
10.4171/196-1/7
https://www.ems-ph.org/doi/10.4171/196-1/7
The Gauss–Bonnet theorem and the geometry of surfaces
Son Lam
Ho
Université de Sherbrooke, Canada
Geometry
This is an expository article on the classical Gauss–Bonnet theorem. Focusing on the intuition behind the ideas, we introduce the concepts of geodesic curvature, Gaussian curvature, and tie them together with an informal proof of the theorem.
113
123
1
10.4171/196-1/8
https://www.ems-ph.org/doi/10.4171/196-1/8
On the non-existence of a perfect map from the 2-sphere to the Euclidean plane
Charalampos
Charitos
Agricultural University of Athens, Greece
Ioannis
Papadoperakis
Agricultural University of Athens, Greece
Geometry
We present a proof of a theorem of Euler asserting that there does not exist any perfect map from the 2-sphere to the Euclidean plane. The meaning of the word „perfect" is explained. Euler's result is part of his broad work on cartography, whose aim was to find representations on the Euclidean plane, with the least possible distorsion, of regions of the sphere. The word „distorsion," in the context we consider, is explained. The ideas in the proof presented here are applied to the study of other similar problems.
125
134
1
10.4171/196-1/9
https://www.ems-ph.org/doi/10.4171/196-1/9
Area preserving maps from the sphere to the Euclidean plane
Charalampos
Charitos
Agricultural University of Athens, Greece
Geometry
We study area-preserving projections from regions of the sphere to the Euclidean plane, in the tradition of Euler. The question is related to Euler's investigations of the problem of cartography, that is, drawing the best possible geographical maps. The distortion from conformality of the projections we study is compared with that of Lambert's cylindrical equal area projection.
135
150
1
10.4171/196-1/10
https://www.ems-ph.org/doi/10.4171/196-1/10
Area and volume in non-Euclidean geometry
Nikolay
Abrosimov
Sobolev Institute of Mathematics, and Novosibirsk State University, Novosibirsk, Russian Federation
Alexander
Mednykh
Sobolev Institute of Mathematics and Novosibirsk State University, Novosibirsk, Russian Federation
Geometry
We give an overview of old and recent results on area and volume in hyperbolic and spherical geometries. First, we present the known results about Heron's and Ptolemy's theorems. Then we present non-Euclidean analogues of Brahmagupta's theorem for a cyclic quadrilateral. We produce also hyperbolic and spherical versions of Bretschneider's formula for the area of a quadrilateral. We give hyperbolic and spherical analogues of Casey's theorem which is a generalization of Ptolemy's equation. We give a short historical review of volume calculations for non-Euclidean polyhedra. Then we concentrate on recent results concerning Seidel's problem on the volume of an ideal tetrahedron, Sforza's formula for a compact tetrahedron in $\mathbb{H}^3$ or $\mathbb{S}^3$ and volumes of non-Euclidean octahedra with symmetries.
151
189
1
10.4171/196-1/11
https://www.ems-ph.org/doi/10.4171/196-1/11
Statics and kinematics of frameworks in Euclidean and non-Euclidean geometry
Ivan
Izmestiev
Université de Fribourg, Switzerland
Geometry
Several aspects of the infinitesimal rigidity of frameworks in Euclidean, hyperbolic, and spherical geometry are presented. First, we prove the equivalence of the static and kinematic formulations of infinitesimal rigidity. By means of the projective interpretation of statics (representing forces as bivectors), this allows us to prove the projective invariance of infinitesimal rigidity and to establish a correspondence between infinitesimal motions of a Euclidean framework and of its geodesic image in any other geometry of constant curvature. We also describe the Maxwell–Cremona correspondence between equilibrium loads and polyhedral lifts, both for Euclidean and for non-Euclidean frameworks.
191
233
1
10.4171/196-1/12
https://www.ems-ph.org/doi/10.4171/196-1/12
Contributions to non-Euclidean geometry I
Eduard
Study
Universität Bonn, Germany
Geometry
237
251
1
10.4171/196-1/13
https://www.ems-ph.org/doi/10.4171/196-1/13
Notes on Eduard Study’s paper “Contributions to non-Euclidean geometry I”
Annette
A’Campo-Neuen
Universität Basel, Switzerland
Athanase
Papadopoulos
Université de Strasbourg, France
Geometry
This is a commentary on the paper "Contributions to non-Euclidean geometry I" by Eduard Study, published in 1907, in which the author lays the foundations of the exterior geometry of hyperbolic space and what is called today de Sitter geometry.
253
262
1
10.4171/196-1/14
https://www.ems-ph.org/doi/10.4171/196-1/14
Spherical and hyperbolic conics
Ivan
Izmestiev
Université de Fribourg, Switzerland
Geometry
This is a survey of metric properties of spherical and hyperbolic conics, mainly based on works of Chasles and Story. A spherical conic is the intersection of the sphere with a quadratic cone; similarly, a hyperbolic conic is the intersection of the Beltrami–Cayley–Klein disk with an affine conic. Non-Euclidean conics have numerous metric properties similar to those of Euclidean conics. The presence of absolute polarity makes the non-Euclidean case richer than the Euclidean one.
262
320
1
10.4171/196-1/15
https://www.ems-ph.org/doi/10.4171/196-1/15
Spherical, hyperbolic, and other projective geometries: convexity, duality, transitions
François
Fillastre
Université de Cergy-Pontoise, France
Andrea
Seppi
Université du Luxembourg, Esch-sur-Alzette, Luxembourg
Geometry
We give an elementary projective geometry presentation of the classical Riemannian model spaces (elliptic and hyperbolic spaces) and of the classical Lorentzian model spaces (de Sitter and anti-de Sitter spaces). We also present some relevant degenerate model spaces (Euclidean and co-Euclidean spaces, Lorentzian Minkowski and co-Minkowski spaces), and geometric transitions. An emphasis is given to dimensions 2 and 3, convex subsets, duality, and geometric transitions between the spaces.
321
409
1
10.4171/196-1/16
https://www.ems-ph.org/doi/10.4171/196-1/16
Hermitian trigonometry
Boumediene
Et-Taoui
Université de Haute Alsace, Mulhouse, France
Geometry
The purpose of this chapter is to survey Hermitian trigonometry, a subject that was studied at different times by several authors. These authors introduced invariants for triangles in the complex projective plane and, by different methods, they established the trigonometric laws which turn out to be two sine laws and a cosine law. In this chapter, we present and compare these laws. We shall also see that those of usual spherical trigonometry can be easily deduced from them. It turns out that one of the sine laws does not appear in the works of Blaschke and Terheggen who were the first to study Hermitian trigonometry. We show that their method directly leads to this law. In addition, we define a new triangle invariant $\alpha$ and show that this invariant and two other essential invariants defined by Blaschke and Terheggen and denoted by $\omega$ and $\Omega$ satisfy a simple equation: $ \Omega=-\alpha-2\omega\ +(2k+1)\pi$, $k \in \mathbb{Z}$.
413
425
1
10.4171/196-1/17
https://www.ems-ph.org/doi/10.4171/196-1/17
A theorem on equiareal triangles with a fixed base
Victor
Pambuccian
Arizona State University, Phoenix, USA
Geometry
The statement: "Given two fixed points, $A$ and $C$, the locus of the midpoints of $AB$ and $CB$, when $B$ varies such that the area of triangle $ABC$ is constant, consists of two lines symmetric with respect to $AC$" is shown to be provable in very weak geometries, that is, Bachmann's non-elliptic metric planes in which every pair of points has a midpoint.
427
437
1
10.4171/196-1/18
https://www.ems-ph.org/doi/10.4171/196-1/18
Function Spaces with Dominating Mixed Smoothness
Hans
Triebel
Friedrich-Schiller Universität Jena, Germany
Functional analysis
Fourier analysis
46-02, 46E35, 42C40, 42B35, 41A55
Functional analysis
Function spaces, dominating mixed smoothness, spaces on domains, wavelets, Faber frames, Haar frames
The first part of this book is devoted to function spaces in Euclidean $n$-space with dominating mixed smoothness. Some new properties are derived and applied in the second part where weighted spaces with dominating mixed smoothness in arbitrary bounded domains in Euclidean $n$-space are introduced and studied. This includes wavelet frames, numerical integration and discrepancy, measuring the deviation of sets of points from uniformity. These notes are addressed to graduate students and mathematicians having a working knowledge of basic elements of the theory of function spaces, especially of Besov–Sobolev type. In particular, it will be of interest for researchers dealing with approximation theory, numerical integration and discrepancy.
1
31
2019
978-3-03719-195-8
978-3-03719-695-3
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/195
http://www.ems-ph.org/doi/10.4171/195
EMS Series of Lectures in Mathematics
2523-5176
2523-5184
From the Vlasov–Maxwell–Boltzmann System to Incompressible Viscous Electro-magneto-hydrodynamics
Volume 1
Diogo
Arsénio
Université Paris Diderot, France
Laure
Saint-Raymond
École Normale Supérieure, Lyon, France
Fluid mechanics
Partial differential equations
Statistical mechanics, structure of matter
76P05, 76W05, 82C40, 35B25
Fluid mechanics
Plasma, magneto-hydro-dynamics, fluid limits, kinetic theory, entropy method, moment method, hypoellipticity, electromagetic waves, Ohm’s law
The Vlasov–Maxwell–Boltzmann system is a microscopic model to describe the dynamics of charged particles subject to self-induced electromagnetic forces. At the macroscopic scale, in the incompressible viscous fluid limit the evolution of the plasma is governed by equations of Navier–Stokes–Fourier type, with some electromagnetic forcing that may take on various forms depending on the number of species and on the strength of the interactions. From the mathematical point of view, these models have very different behaviors. Their analysis therefore requires various mathematical methods which this book aims to present in a systematic, painstaking, and exhaustive way. The first part of this work is devoted to the systematic formal analysis of viscous hydrodynamic limits of the Vlasov–Maxwell–Boltzmann system, leading to a precise classification of physically relevant models for viscous incompressible plasmas, some of which have not previously been described in the literature. In the second part, the convergence results are made precise and rigorous, assuming the existence of renormalized solutions for the Vlasov–Maxwell–Boltzmann system. The analysis is based essentially on the scaled entropy inequality. Important mathematical tools are introduced, with new developments used to prove these convergence results (Chapman–Enskog-type decomposition and regularity in the $v$ variable, hypoelliptic transfer of compactness, analysis of high frequency time oscillations, and more). The third and fourth parts (which will be published in a second volume) show how to adapt the arguments presented in the conditional case to deal with a weaker notion of solutions to the Vlasov–Maxwell–Boltzmann system, the existence of which is known.
3
31
2019
978-3-03719-193-4
978-3-03719-693-9
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/193
https://www.ems-ph.org/doi/10.4171/193
EMS Monographs in Mathematics
2523-5192
2523-5206
The Shock Development Problem
Demetrios
Christodoulou
ETH Zürich, Switzerland
Partial differential equations
Fluid mechanics
35L67; 35L04, 35L65, 76L05, 76N15
Differential equations
Nonlinear hyperbolic partial differential equations, free boundary problems, mechanics of compressible fluids, development of shocks
This monograph addresses the problem of the development of shocks in the context of the Eulerian equations of the mechanics of compressible fluids. The mathematical problem is that of an initial-boundary value problem for a nonlinear hyperbolic system of partial differential equations with a free boundary and singular initial conditions. The free boundary is the shock hypersurface and the boundary conditions are jump conditions relative to a prior solution, conditions following from the integral form of the mass, momentum and energy conservation laws. The prior solution is provided by the author‘s previous work which studies the maximal classical development of smooth initial data. New geometric and analytic methods are introduced to solve the problem. Geometry enters as the acoustical structure, a Lorentzian metric structure defined on the spacetime manifold by the fluid. This acoustical structure interacts with the background spacetime structure. Reformulating the equations as two coupled first order systems, the characteristic system, which is fully nonlinear, and the wave system, which is quasilinear, a complete regularization of the problem is achieved. Geometric methods also arise from the need to treat the free boundary. These methods involve the concepts of bi-variational stress and of variation fields. The main new analytic method arises from the need to handle the singular integrals appearing in the energy identities. Shocks being an ubiquitous phenomenon, occuring also in magnetohydrodynamics, nonlinear elasticity, and the electrodynamics of noninear media, the methods developed in this monograph are likely to be found relevant in these fields as well.
1
31
2019
978-3-03719-192-7
978-3-03719-692-2
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/192
http://www.ems-ph.org/doi/10.4171/192
EMS Monographs in Mathematics
2523-5192
2523-5206
Estimates for Differential Operators in Half-space
Translated from the German by Darya Apushkinskaya
Igor
Gel'man
Akko, Israel
Vladimir
Maz'ya
Linköping University, Sweden, and University of Liverpool, UK
Partial differential equations
35G05, 35S05
Differential equations
Differential operators with constant coefficients, differential operators in a half-space, pseudo-differential operators, domination of differential operators, boundary traces, maximal operator, estimates for Lame system, estimates for Stokes system
Inequalities for differential operators play a fundamental role in the modern theory of partial differential equations. Among the numerous applications of such inequalities are existence and uniqueness theorems, error estimates for numerical approximations of solutions and for residual terms in asymptotic formulas, as well as results on the structure of the spectrum. The inequalities cover a wide range of differential operators, boundary conditions and norms of the corresponding function spaces. The book focuses on estimates up to the boundary of a domain. It contains a great variety of inequalities for differential and pseudodifferential operators with constant coefficients. Results of final character are obtained, without any restrictions on the type of differential operators. Algebraic necessary and sufficient conditions for the validity of the corresponding a priori estimates are presented. General criteria are systematically applied to particular types of operators found in classical equations and systems of mathematical physics (such as Lame’s system of static elasticity theory or the linearized Navier–Stokes system), Cauchy–Riemann’s operators, Schrödinger operators, among others. The well-known results of Aronszajn, Agmon–Douglis–Nirenberg and Schechter fall into the general scheme, and sometimes are strengthened. The book will be interesting and useful to a wide audience, including graduate students and specialists in the theory of differential equations.
1
31
2019
978-3-03719-191-0
978-3-03719-691-5
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/191
http://www.ems-ph.org/doi/10.4171/191
EMS Tracts in Mathematics
31
Boundary Behavior of Solutions to Elliptic Equations in General Domains
Vladimir
Maz'ya
Linköping University, Sweden and University of Liverpool, UK
Partial differential equations
Potential theory
35J40, 31B15, 31B25
Differential equations
Wiener test, higher order elliptic equations, elasticity systems, Zaremba problem, weighted positivity, capacity
The present book is a detailed exposition of the author and his collaborators’ work on boundedness, continuity, and differentiability properties of solutions to elliptic equations in general domains, that is, in domains that are not a priori restricted by assumptions such as “piecewise smoothness” or being a “Lipschitz graph”. The description of the boundary behavior of such solutions is one of the most difficult problems in the theory of partial differential equations. After the famous Wiener test, the main contributions to this area were made by the author. In particular, necessary and sufficient conditions for the validity of imbedding theorems are given, which provide criteria for the unique solvability of boundary value problems of second and higher order elliptic equations. Another striking result is a test for the regularity of a boundary point for polyharmonic equations. The book will be interesting and useful for a wide audience. It is intended for specialists and graduate students working in the theory of partial differential equations.
9
30
2018
978-3-03719-190-3
978-3-03719-690-8
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/190
http://www.ems-ph.org/doi/10.4171/190
EMS Tracts in Mathematics
30
Geometric and Topological Aspects of Coxeter Groups and Buildings
Anne
Thomas
The University of Sydney, Australia
Group theory and generalizations
Primary: 20F55; secondary: 20E42, 51E24, 57M07
Groups + group theory
Coxeter groups, buildings, Davis complexes
Coxeter groups are groups generated by reflections, and they appear throughout mathematics. Tits developed the general theory of Coxeter groups in order to develop the theory of buildings. Buildings have interrelated algebraic, combinatorial and geometric structures, and are powerful tools for understanding the groups which act on them. These notes focus on the geometry and topology of Coxeter groups and buildings, especially nonspherical cases. The emphasis is on geometric intuition, and there are many examples and illustrations. Part I describes Coxeter groups and their geometric realisations, particularly the Davis complex, and Part II gives a concise introduction to buildings. This book will be suitable for mathematics graduate students and researchers in geometric group theory, as well as algebra and combinatorics. The assumed background is basic group theory, including group actions, and basic algebraic topology, together with some knowledge of Riemannian geometry.
5
31
2018
978-3-03719-189-7
978-3-03719-689-2
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/189
http://www.ems-ph.org/doi/10.4171/189
Zurich Lectures in Advanced Mathematics
A Brief Introduction to Spectral Graph Theory
Bogdan
Nica
McGill University, Montreal, Canada
Combinatorics
Number theory
Linear and multilinear algebra; matrix theory
Primary: 05-01, 05C50; secondary: 05C25, 11T24, 15A42
Combinatorics + graph theory
Adjacency eigenvalues of graphs, Laplacian eigenvalues of graphs, Cayley graphs, algebraic graphs over finite fields, character sums
Spectral graph theory starts by associating matrices to graphs – notably, the adjacency matrix and the Laplacian matrix. The general theme is then, firstly, to compute or estimate the eigenvalues of such matrices, and secondly, to relate the eigenvalues to structural properties of graphs. As it turns out, the spectral perspective is a powerful tool. Some of its loveliest applications concern facts that are, in principle, purely graph theoretic or combinatorial. This text is an introduction to spectral graph theory, but it could also be seen as an invitation to algebraic graph theory. The first half is devoted to graphs, finite fields, and how they come together. This part provides an appealing motivation and context of the second, spectral, half. The text is enriched by many exercises and their solutions. The target audience are students from the upper undergraduate level onwards. We assume only a familiarity with linear algebra and basic group theory. Graph theory, finite fields, and character theory for abelian groups receive a concise overview and render the text essentially self-contained.
5
31
2018
978-3-03719-188-0
978-3-03719-688-5
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/188
http://www.ems-ph.org/doi/10.4171/188
EMS Textbooks in Mathematics
An Introduction to Kac–Moody Groups over Fields
Timothée
Marquis
Université Catholique de Louvain, Louvain-la-Neuve, Belgium
Group theory and generalizations
20G44, 20E42, 17B67
Groups + group theory
Kac–Moody groups, Kac–Moody algebras, infinite-dimensional Lie theory, highest-weight modules, semisimple algebraic groups, loop groups, affine group schemes, Coxeter groups, buildings, BN pairs, Tits systems, root group data
The interest for Kac–Moody algebras and groups has grown exponentially in the past decades, both in the mathematical and physics communities, and with it also the need for an introductory textbook on the topic. The aims of this book are twofold: - to offer an accessible, reader-friendly and self-contained introduction to Kac–Moody algebras and groups; - to clean the foundations and to provide a unified treatment of the theory. The book starts with an outline of the classical Lie theory, used to set the scene. Part II provides a self-contained introduction to Kac–Moody algebras. The heart of the book is Part III, which develops an intuitive approach to the construction and fundamental properties of Kac–Moody groups. It is complemented by two appendices, respectively offering introductions to affine group schemes and to the theory of buildings. Many exercises are included, accompanying the readers throughout their journey. The book assumes only a minimal background in linear algebra and basic topology, and is addressed to anyone interested in learning about Kac–Moody algebras and/or groups, from graduate (master) students to specialists.
6
15
2018
978-3-03719-187-3
978-3-03719-687-8
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/187
http://www.ems-ph.org/doi/10.4171/187
EMS Textbooks in Mathematics
Non-Linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis
The Helge Holden Anniversary Volume
Fritz
Gesztesy
Baylor University, Waco, USA
Harald
Hanche-Olsen
The Norwegian University of Science and Technology, Trondheim, Norway
Espen
Jakobsen
The Norwegian University of Science and Technology, Trondheim, Norway
Yurii
Lyubarskii
The Norwegian University of Science and Technology, Trondheim, Norway
Nils Henrik
Risebro
University of Oslo, Norway
Kristian
Seip
Norwegian University of Science and Technology, Trondheim, Norway
Partial differential equations
Linear and multilinear algebra; matrix theory
Dynamical systems and ergodic theory
Fourier analysis
Primary: 15B52, 35J10, 35L65, 35Q41, 35Q51, 35Q53, 37K10, 42B20, 46N20, 46N30, 46T12, 47B36, 47F05, 60H20, 68N30, 76S05; Secondary: 33C45, 35A01, 35A02, 35L80, 37D45, 39A12, 47A10, 47N20, 47N30, 60B20
Differential equations
Infinite-dimensional analysis, partial differential equations, hyperbolic conservation laws, stochastic analysis, spectral theory, discrete evolution, completely integrable systems, random matrix theory, chaotic dynamics
This volume is dedicated to Helge Holden on the occasion of his 60th anniversary. It collects contributions by numerous scientists with expertise in non-linear partial differential equations (PDEs), mathematical physics, and stochastic analysis, reflecting to a large degree Helge Holden’s longstanding research interests. Accordingly, the problems addressed in the contributions deal with a large range of topics, including, in particular, infinite-dimensional analysis, linear and nonlinear PDEs, stochastic analysis, spectral theory, completely integrable systems, random matrix theory, and chaotic dynamics and sestina poetry. They represent to some extent the lectures presented at the conference Non-linear PDEs, Mathematical Physics and Stochastic Analysis, held at NTNU, Trondheim, July 4–7, 2016 (https://wiki.math.ntnu.no/holden60). The mathematical tools involved draw from a wide variety of techniques in functional analysis, operator theory, and probability theory. This collection of research papers will be of interest to any active scientist working in one of the above mentioned areas.
5
31
2018
978-3-03719-186-6
978-3-03719-686-1
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/186
http://www.ems-ph.org/doi/10.4171/186
EMS Series of Congress Reports
2523-515X
2523-5168
Optimal control of forward-backward stochastic Volterra equations
Nacira
Agram
University of Oslo, Norway
Bernt
Øksendal
University of Oslo, Norway
Samia
Yakhlef
University of Biskra, Algeria
Partial differential equations
We study the problem of optimal control of a coupled system of forward-backward stochastic Volterra equations. We use Hida–Malliavin calculus to prove a sufficient and a necessary maximum principle for the optimal control of such systems. Existence and uniqueness of backward stochastic Volterra integral equations are proved. As an application of our methods, we solve a recursive utility optimisation problem in a financial model with memory.
3
35
1
10.4171/186-1/1
http://www.ems-ph.org/doi/10.4171/186-1/1
A unified approach to infinite dimensional integrals of probabilistic and oscillatory type with applications to Feynman path integrals
Sergio
Albeverio
Universität Bonn, Germany
Sonia
Mazzucchi
Università di Trento, Povo (Trento), Italy
Partial differential equations
An unified approach to infinite dimensional integration in terms of linear continuous functional is presented, including the cases of both probabilistic and oscillatory integrals. Applications to the theory of Feynman path integrals and to the study of high-order heat-type equations are also presented.
37
53
1
10.4171/186-1/2
http://www.ems-ph.org/doi/10.4171/186-1/2
The numbers lead a dance. Mathematics of the Sestina
Alan
Champneys
University of Bristol, UK
Poul
Hjorth
Technical University of Denmark, Kgs. Lyngby, Denmark
Harry
Man
Oxford Brookes University, UK
Partial differential equations
Sestinas are poems of 39 lines comprising six verses of six lines each, and a three line final verse or ‘envoi’. The structure of the sestina is built around word repetition rather than strict rhyme. Each verse uses the same set line ending words, but in a permuted order. The form of the permutation is highly specific, and is equivalent to iteration of the tent map. This paper considers for which number $N$ of verses, other than 6, can a sestina-like poem be formed. That is, which $N$ will the prescribed permutation lead to a poem of $N$ verses where no two verses have the same order of their end words. In so doing, a link is found between permutation groups, chaotic dynamics, and Cunningham numbers.
55
71
1
10.4171/186-1/3
http://www.ems-ph.org/doi/10.4171/186-1/3
Compensated compactness in Banach spaces and weak rigidity of isometric immersions of manifolds
Gui-Qiang
Chen
Oxford University, UK
Siran
Li
Rice University, Houston, USA
Partial differential equations
We present a compensated compactness theorem in Banach spaces established recently, whose formulation is originally motivated by the weak rigidity problem for isometric immersions of manifolds with lower regularity. As a corollary, a geometrically intrinsic div–curl lemma for tensor fields on Riemannian manifolds is obtained. Then we show how this intrinsic div–curl lemma can be employed to establish the global weak rigidity of the Gauss–Codazzi–Ricci equations, the Cartan formalism, and the corresponding isometric immersions of Riemannian submanifolds.
73
95
1
10.4171/186-1/4
http://www.ems-ph.org/doi/10.4171/186-1/4
The initial-boundary-value problem for an Ostrovsky–Hunter type equation
Giuseppe Maria
Coclite
Politecnico di Bari, Italy
Lorenzo
di Ruvo
Politecnico di Bari, Italy
Kenneth Hvistendahl
Karlsen
University of Oslo, Norway
Partial differential equations
We consider an Ostrovsky–Hunter type equation. We prove the well-posedness of the entropy solution for the non-homogeneous initial boundary value problem. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method.
97
109
1
10.4171/186-1/5
http://www.ems-ph.org/doi/10.4171/186-1/5
Modeling crowd dynamics through hyperbolic – elliptic equations
Rinaldo
Colombo
Università degli Studi di Brescia, Italy
Maria
Gokieli
University of Warsaw, Poland
Massimiliano
Rosini
Maria Curie-Skłodowska University (UMCS), Lublin, Poland
Partial differential equations
Inspired by the works of Hughes [22, 23], we formalize and prove the well-posedness of a hyperbolic–elliptic system whose solutions describe the dynamics of a moving crowd. The resulting model is here shown to be well-posed and the time of evacuation from a bounded environment is proved to be finite. This model also provides a microscopic description of the individuals’ behaviors.
111
128
1
10.4171/186-1/6
http://www.ems-ph.org/doi/10.4171/186-1/6
On the well-posedness of solutions with finite energy for nonlocal equations of porous medium type
Félix
del Teso
The Norwegian University of Science and Technology (NTNU), Trondheim, Norway
Jørgen
Endal
The Norwegian University of Science and Technology (NTNU), Trondheim, Norway
Espen
Jakobsen
The Norwegian University of Science and Technology (NTNU), Trondheim, Norway
Partial differential equations
We study well-posedness and equivalence of different notions of solutions with finite energy for nonlocal porous medium type equations of the form $$\partial_t u – A \varphi (u) = 0.$$ These equations are possibly degenerate nonlinear diffusion equations with a general nondecreasing continuous nonlinearity $\varphi$, and the largest class of linear symmetric nonlocal diffusion operators $A$ considered so far. The operators are defined from a bilinear energy form $\mathcal E$ and may be degenerate and have some $x$-dependence. The fractional Laplacian, symmetric finite differences, and any generator of symmetric pure jump Lévy processes are included. The main results are (i) an Oleĭnik type uniqueness result for energy solutions; (ii) an existence (and uniqueness) result for distributional solutions with finite energy; and (iii) equivalence between the two notions of solution, and as a consequence, new wellposedness results for both notions of solutions. We also obtain quantitative energy and related $L^p$-estimates for distributional solutions. Our uniqueness results are given for a class of functions defined from test functions by completion in a certain topology. We study rigorously several cases where this space coincides with standard function spaces. In particular, for operators comparable to fractional Laplacians, we show that this space is a parabolic homogeneous fractional Sobolev space.
129
167
1
10.4171/186-1/7
http://www.ems-ph.org/doi/10.4171/186-1/7
On the spectrum of leaky surfaces with a potential bias
Pavel
Exner
Czech Academy of Sciences, Řež near Prague, Czechia, and Czech Technical University, Prague, Czechia
Partial differential equations
We discuss operators of the type $H = –\Delta + V(x) – \alpha \delta (x–\Sigma)$ with an attractive interaction, $\alpha > 0$ in $L^2(\mathbb R^3)$, where $\Sigma$ is an infinite surface, asymptotically planar and smooth outside a compact, dividing the space into two regions, of which one is supposed to be convex, and $VB$ is a potential bias being a positive constant $V_0$ in one of the regions and zero in the other. We find the essential spectrum and ask about the existence of the discrete one with a particular attention to the critical case, $V_0 = \alpha^2$. We show that $\sigma_{\mathrm {disc}} (H)$ is then empty if the bias is supported in the ‘exterior’ region, while in the opposite case isolated eigenvalues may exist.
169
181
1
10.4171/186-1/8
http://www.ems-ph.org/doi/10.4171/186-1/8
On the decay of almost periodic solutions of anisotropic degenerate parabolic-hyperbolic equations
Hermano
Frid
Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil
Partial differential equations
We prove the well-posedness and decay of Besicovitch almost periodic solutions for nonlinear degenerate anisotropic hyperbolic-parabolic equations. The decay property is proven for the case where the diffusion term is given by a non-degenerate nonlinear $d’’ \times d’’$ diffusion matrix and the complementary $d’$ components of flux-function form a non-degenerate flux in $\mathbb R^{d’}$, with $d’+ d’’ = d$. For this special case we also prove that the strong trace property at the initial time holds, which allows, in particular, to require the assumption of the initial data only in a weak sense, and gives the continuity in time of the solution with values in $L^1_{\mathrm {loc}} (\mathbb R^d)$). So far, for the decay property, we need also to require that the bounded Besicovitch almost periodic initial function can be approximated in the Besicovitch norm by almost periodic functions whose $\epsilon$-inclusion intervals $l_\epsilon$ satisfy $l_\epsilon | \mathrm {log}\: \epsilon |^1/2 \to 0$ as $\epsilon \to 0$. This includes, in particular, generalized limit periodic functions, that is, limits in the Besicovitch norm of purely periodic functions.
183
205
1
10.4171/186-1/9
http://www.ems-ph.org/doi/10.4171/186-1/9
Factorizations and Hardy–Rellich-type inequalities
Fritz
Gesztesy
Baylor University, Waco, USA
Lance
Littlejohn
Baylor University, Waco, USA
Partial differential equations
The principal aim of this note is to illustrate how factorizations of singular, even-order partial differential operators yield an elementary approach to classical inequalities of Hardy–Rellich-type. More precisely, introducing the two-parameter $n$-dimensional homogeneous scalar differential expressions $T_{\alpha,\beta} := - \Delta + \alpha |x|^{-2} x \cdot \nabla + \beta |x|^{-2}$, $\alpha, \beta \in \mathbb R$, $x \in \mathbb R^n \setminus \{0\}$, $n \in \mathbb N$, $n \geq 2$, and its formal adjoint, denoted by $T_{\alpha,\beta}^+$, we show that nonnegativity of $T_{\alpha,\beta}^+ T_{\alpha,\beta}$ on $C_0^{\infty}(\mathbb R^n \setminus \{0\})$ implies the fundamental inequality (*) \begin{equation}\tag{$*$}\label{0.1} \begin{aligned} \int_{\mathbb R^n} [(\Delta f)(x)]^2 \, d^n x & \geq [(n - 4) \alpha - 2 \beta] \int_{\mathbb R^n} |x|^{-2} |(\nabla f)(x)|^2 \, d^n x \\ & \quad - \alpha (\alpha - 4) \int_{\mathbb R^n} |x|^{-4} |x \cdot (\nabla f)(x)|^2 \, d^n x \\ & \quad + \beta [(n - 4) (\alpha - 2) - \beta] \int_{\mathbb R^n} |x|^{-4} |f(x)|^2 \, d^n x,\\ &&\llap {f \in C^{\infty}_0(\mathbb R^n \setminus \{0\}).} \end{aligned} \end{equation} A particular choice of values for $\alpha$ and $\beta$ in (*) yields known Hardy–Rellich-type inequalities, including the classical Rellich inequality and an inequality due to Schmincke. By locality, these inequalities extend to the situation where $\mathbb R^n$ is replaced by an arbitrary open set $\Omega \subseteq \mathbb R^n$ for functions $f \in C^{\infty}_0(\Omega \setminus \{0\})$. Perhaps more importantly, we will indicate that our method, in addition to being elementary, is quite flexible when it comes to a variety of generalized situations involving the inclusion of remainder terms and higher-order operators.
207
226
1
10.4171/186-1/10
http://www.ems-ph.org/doi/10.4171/186-1/10
Symmetries and multipeakon solutions for the modified two-component Camassa–Holm system
Katrin
Grunert
The Norwegian University of Science and Technology (NTNU), Trondheim, Norway
Xavier
Raynaud
SINTEF, Oslo, Norway, and The Norwegian University of Science and Technology (NTNU), Trondheim, Norway
Partial differential equations
Compared with the two-component Camassa–Holm system, the modified two-component Camassa–Holm system introduces a regularized density which makes possible the existence of solutions of lower regularity, and in particular of multipeakon solutions. In this paper, we derive a new pointwise invariant for the modified two-component Camassa–Holm system. The derivation of the invariant uses directly the symmetry of the system, following the classical argument of Noether’s theorem. The existence of the multipeakon solutions can be directly inferred from this pointwise invariant. This derivation shows the strong connection between symmetries and the existence of special solutions. The observation also holds for the scalar Camassa–Holm equation and, for comparison, we have also included the corresponding derivation. Finally, we compute explicitly the solutions obtained for the peakon-antipeakon case. We observe the existence of a periodic solution which has not been reported in the literature previously. This case shows the attractive effect that the introduction of an elastic potential can have on the solutions.
227
260
1
10.4171/186-1/11
http://www.ems-ph.org/doi/10.4171/186-1/11
Vanishing viscosity solutions of Riemann problems for models of polymer flooding
Graziano
Guerra
Università degli Studi di Milano-Bicocca, Italy
Wen
Shen
The Pennsylvania State University, University Park, USA
Partial differential equations
We consider the solutions of Riemann problems for polymer flooding models. In a suitable Lagrangian coordinate the systems take a triangular form, where the equation for thermodynamics is decoupled from the hydrodynamics, leading to the study of scalar conservation laws with discontinuous flux functions. We prove three equivalent admissibility conditions for shocks for scalar conservation laws with discontinuous flux. Furthermore, we show that a variation of minimum path of [10] proposed in [18] is the vanishing viscosity limit of a partially viscous model with viscosity only in the hydro-dynamics.
261
285
1
10.4171/186-1/12
http://www.ems-ph.org/doi/10.4171/186-1/12
Efficient computation of all speed flows using an entropy stable shock-capturing space-time discontinuous Galerkin method
Andreas
Hiltebrand
ANSYS Switzerland, Zürich, Switzerland
Siddhartha
Mishra
ETH Zürich, Switzerland and University of Oslo, Norway
Partial differential equations
We present a shock-capturing space-time discontinuous Galerkin method to approximate all speed flows modeled by systems of conservation laws with multiple time scales. The method provides a very general and computationally efficient framework for approximating such systems on account of its ability to incorporate large time steps. Numerical examples ranging from computing the incompressible limit (robustness with respect to Mach number) of the Euler equations to accelerating convergence to steady state are presented for illustrating the method.
287
318
1
10.4171/186-1/13
http://www.ems-ph.org/doi/10.4171/186-1/13
Dispersion estimates for spherical Schrödinger equations with critical angular momentum
Markus
Holzleitner
Universität Wien, Austria
Aleksey
Kostenko
Universität Wien, Austria
Gerald
Teschl
Universität Wien, Austria and Erwin Schrödinger Institut, Wien, Austria
Partial differential equations
We derive a dispersion estimate for one-dimensional perturbed radial Schrödinger operators, where the angular momentum takes the critical value $l = –1/2$. We also derive several new estimates for solutions of the underlying differential equation and investigate the behavior of the Jost function near the edge of the continuous spectrum.
319
347
1
10.4171/186-1/14
http://www.ems-ph.org/doi/10.4171/186-1/14
Sixty years of moments for random matrices
Werner
Kirsch
FernUniversität Hagen, Germany
Thomas
Kriecherbauer
Universität Bayreuth, Germany
Partial differential equations
This is an elementary review, aimed at non-specialists, of results that have been obtained for the limiting distribution of eigenvalues and for the operator norms of real symmetric random matrices via the method of moments. This method goes back to a remarkable argument of Eugene Wigner some sixty years ago which works best for independent matrix entries, as far as symmetry permits, that are all centered and have the same variance. We then discuss variations of this classical result for ensembles for which the variance may depend on the distance of the matrix entry to the diagonal, including in particular the case of band random matrices, and/or for which the required independence of the matrix entries is replaced by some weaker condition. This includes results on ensembles with entries from Curie–Weiss random variables or from sequences of exchangeable random variables that have been obtained quite recently.
349
379
1
10.4171/186-1/15
http://www.ems-ph.org/doi/10.4171/186-1/15
Bound states of Schrödinger type operators with Heisenberg sub-Laplacian
Ari
Laptev
Imperial College London, UK
Andrei
Velicu
Imperial College London, UK
Partial differential equations
Using the technique from [8] we find a new constant in a Cwikel–Lieb–Rozenblum type inequality that estimate the number of negative eigenvalues of a Schrödinger operator involving the Heisenberg sub-Laplacian with a potential that is proportional to the characteristic function of a measurable set.
381
387
1
10.4171/186-1/16
http://www.ems-ph.org/doi/10.4171/186-1/16
On Holden’s seven guidelines for scientific computing and development of open-source community software
Knut-Andreas
Lie
SINTEF, Oslo, Norway and NTNU, Trondheim, Norway
Partial differential equations
Two decades ago, Helge Holden proposed seven guidelines to improve the way new achievements and results in scientific computing were presented, evaluated, and compared in contemporary scientific literature. In this essay, written as a tribute to Helge on his 60th birthday, I revisit the guidelines and point out why they are still valid today seen from my perspective, working as a contract researcher at the interface between mathematics and applications in industry. Developing new computational methods usually involves a lot of experimental programming. Over the past decade, my research group has developed an open-source community code that today has hundreds of users worldwide. I discuss some considerations that have gone into this development and present a few lessons learned. Moreover, based on this experience, as well as from development of professional software for our clients, I present advice on how you can be more productive in your experimental programming and increase the impact of your scientific results.
389
422
1
10.4171/186-1/17
http://www.ems-ph.org/doi/10.4171/186-1/17
Sharp uniqueness results for discrete evolutions
Yurii
Lyubarskii
The Norwegian University of Science and Technology (NTNU), Trondheim, Norway
Eugenia
Malinnikova
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
Partial differential equations
We prove sharp uniqueness results for a wide class of one-dimensional discrete evolutions. The proof is based on a construction from the theory of complex Jacobi matrices combined with growth estimates of entire functions.
423
436
1
10.4171/186-1/18
http://www.ems-ph.org/doi/10.4171/186-1/18
Spatial analyticity of solutions to nonlinear dispersive PDE
Sigmund
Selberg
University of Bergen, Norway
Partial differential equations
Given a nonlinear dispersive PDE, for example the KdV equation, we consider the Cauchy problem with real analytic initial data. For data with a uniform radius of analyticity, we are interested in obtaining lower bounds on the radius of analyticity at later times. A rather general approach to this problem is presented, based on Bourgain’s Fourier restriction norm method. Applications to the KdV equation (periodic and non-periodic) and the Dirac–Klein–Gordon equations are discussed.
437
454
1
10.4171/186-1/19
http://www.ems-ph.org/doi/10.4171/186-1/19
Local Representation Theory and Simple Groups
Radha
Kessar
City University of London, UK
Gunter
Malle
Universität Kaiserslautern, Germany
Donna
Testerman
EPF Lausanne, Switzerland
Group theory and generalizations
20BXX, 20CXX, 20GXX
Groups + group theory
Finite reductive groups, Deligne-Lusztig varities, Brauer $p$-blocks, local-global conjectures, base size, fixed-point ratios, random walks
The book contains extended versions of seven short lecture courses given during a semester programme on "Local Representation Theory and Simple Groups" held at the Centre Interfacultaire Bernoulli of the EPF Lausanne. These focussed on modular representation theory of finite groups, modern Clifford theoretic methods, the representation theory of finite reductive groups, as well as on various applications of character theory and representation theory, for example to base sizes and to random walks. These lectures are intended to form a good starting point for graduate students and researchers who wish to familiarize themselves with the foundations of the topics covered here. Furthermore they give an introduction to current research directions, including the state of some open problems in the field.
4
30
2018
978-3-03719-185-9
978-3-03719-685-4
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/185
http://www.ems-ph.org/doi/10.4171/185
EMS Series of Lectures in Mathematics
2523-5176
2523-5184
Basic local representation theory
Burkhard
Külshammer
Friedrich-Schiller-Universität Jena, Germany
Group theory and generalizations
1
22
1
10.4171/185-1/1
http://www.ems-ph.org/doi/10.4171/185-1/1
Reduction theorems for some global–local conjectures
Britta
Späth
Bergische Universität Wuppertal, Germany
Group theory and generalizations
23
61
1
10.4171/185-1/2
http://www.ems-ph.org/doi/10.4171/185-1/2
A first guide to the character theory of finite groups of Lie type
Meinolf
Geck
Universität Stuttgart, Germany
Group theory and generalizations
63
106
1
10.4171/185-1/3
http://www.ems-ph.org/doi/10.4171/185-1/3
Lectures on modular Deligne–Lusztig theory
Olivier
Dudas
Université Paris Diderot Paris 7, France
Group theory and generalizations
107
177
1
10.4171/185-1/4
http://www.ems-ph.org/doi/10.4171/185-1/4
Local methods for blocks of finite simple groups
Marc
Cabanes
Université Paris Diderot Paris 7, France
Group theory and generalizations
179
265
1
10.4171/185-1/5
http://www.ems-ph.org/doi/10.4171/185-1/5
Simple groups, fixed point ratios and applications
Timothy
Burness
University of Bristol, UK
Group theory and generalizations
267
322
1
10.4171/185-1/6
http://www.ems-ph.org/doi/10.4171/185-1/6
Applications of character theory of finite simple groups
Martin
Liebeck
Imperial College, London, UK
Group theory and generalizations
323
352
1
10.4171/185-1/7
http://www.ems-ph.org/doi/10.4171/185-1/7
Lectures in Model Theory
Franziska
Jahnke
Universität Münster, Germany
Daniel
Palacín
The Hebrew University of Jerusalem, Israel
Katrin
Tent
Universität Münster, Germany
Mathematical logic and foundations
Combinatorics
Field theory and polynomials
Algebraic geometry
Primary: 03C45, 03C60, 03C98. Secondary: 05E15, 12J20, 12L12, 14E18, 20E18
Mathematical logic
Algebraic geometry
Model theory, stability theory, NIP theories, definably amenable groups, profinite groups, valuation theory, algebraically closed valued fields, motivic integration
Model theory is a thriving branch of mathematical logic with strong connections to other fields of mathematics. Its versatility has recently led to spectacular applications in areas ranging from diophantine geometry, algebraic number theory and group theory to combinatorics. This volume presents lecture notes from a spring school in model theory which took place in Münster, Germany. The notes are aimed at PhD students but should also be accessible to undergraduates with some basic knowledge in model theory. They contain the core of stability theory (Bays, Palacín), two chapters connecting generalized stability theory with group theory (Clausen and Tent, Simon), as well as introductions to the model theory of valued fields (Hils, Jahnke) and motivic integration (Halupczok).
4
30
2018
978-3-03719-184-2
978-3-03719-684-7
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/184
http://www.ems-ph.org/doi/10.4171/184
Münster Lectures in Mathematics
2523-5230
2523-5249
An introduction to stability theory
Daniel
Palacín
The Hebrew University of Jerusalem, Israel
Mathematical logic and foundations
These lecture notes are based on the first section of Pillay's book [3] and they cover fundamental notions of stability theory such as defi nable types, forking calculus and canonical bases, as well as stable groups and homogeneous spaces. The approach followed here is originally due to Hrushovski and Pillay [2], who presented stability from a local point of view. Throughout the notes, some general knowledge of model theory is assumed. I recommend the book of Tent and Ziegler [4] as an introduction to model theory. Furthermore, the texts of Casanovas [1] and Wagner [5] may also be useful to the reader to obtain a di fferent approach to stability theory.
1
27
1
10.4171/184-1/1
http://www.ems-ph.org/doi/10.4171/184-1/1
Geometric stability theory
Martin
Bays
Universität Münster, Germany
Mathematical logic and foundations
These notes cover some of the foundational results of geometric stability theory. We focus on the geometry of minimal sets. The main aim is an account of Hrushovski's result that unimodular (in particular, locally finite or pseudo finite) minimal sets are locally modular; along the way, we discuss the Zilber trichotomy and the group and field confi gurations. We assume the basics of stability theory (forking calculus, U-rank, canonical bases, stable groups and homogeneous spaces), as can be found e.g. in Daniel Palac ín's chapter in this volume [5].
29
58
1
10.4171/184-1/2
http://www.ems-ph.org/doi/10.4171/184-1/2
NIP and definably amenable groups
Pierre
Simon
University of California, Berkeley, USA
Mathematical logic and foundations
This text is an introduction to de finably amenable NIP groups. It is based on a number of papers, mainly [6], [7] and [4]. This subject has two origins, the fi rst one is the theory of stable groups and in particular generic types, which were fi rst defi ned by Poizat (see [12]) and have since played a central role throughout stability theory. Later, part of the theory was generalized to groups in simple theories, where generic types are de ned as types, none of whose translates forks over the empty set.
59
82
1
10.4171/184-1/3
http://www.ems-ph.org/doi/10.4171/184-1/3
Some model theory of profinite groups
Tim
Clausen
Universität Münster, Germany
Katrin
Tent
Universität Münster, Germany
Mathematical logic and foundations
The main purpose of these notes is to give more background and details for the results obtained in [13] which rely heavily on deep results by Lazard, Lubotzky, Mann, du Sautoy and others. At the center is Lazard's purely group theoretic characterization of $p$-adic analytic groups given in [8] (see Section 4.2 below). By Lazard's result a compact topological group is a $p$-adic analytic group if and only if it has an open uniformly powerful pro-$p$ subgroup (see Section 3).
83
118
1
10.4171/184-1/4
http://www.ems-ph.org/doi/10.4171/184-1/4
An introduction to valued fields
Franziska
Jahnke
Universität Münster, Germany
Mathematical logic and foundations
The aim of this chapter is to give a short introduction to the algebra of valued fields and thereby to provide the necessary background for the following two chapters. The material presented here is heavily based on the book "Valued Fields" by Engler and Prestel, as well as (unpublished) lectures given by Jochen Koenigsmann at the University of Oxford in Hilary 2010. Many of the proofs presented are taken from (or at least inspired by) one of these two sources.
119
149
1
10.4171/184-1/5
http://www.ems-ph.org/doi/10.4171/184-1/5
Model theory of valued fields
Martin
Hils
Universität Münster, Germany
Mathematical logic and foundations
This chapter presents a variety of classical results on the model theory of valued fi elds.
151
180
1
10.4171/184-1/6
http://www.ems-ph.org/doi/10.4171/184-1/6
An introduction to motivic integration
Immanuel
Halupczok
Heinrich-Heine-Universität Düsseldorf, Germany
Mathematical logic and foundations
This introduction to motivic integration is aimed at readers who have some base knowledge of model theory of valued fields, as provided e.g. by the notes [9] by Martin Hils in this volume. I will not assume a lot of knowledge about valued fields.
181
202
1
10.4171/184-1/7
http://www.ems-ph.org/doi/10.4171/184-1/7
Linear Forms in Logarithms and Applications
Yann
Bugeaud
Université de Strasbourg, France
Number theory
Primary: 11-02, 11J86, 11D; Secondary: 11B37, 11D25, 11D41, 11D59, 11D61, 11D75, 11D88, 11J25, 11J81, 11J82
Number theory
Baker's theory, linear form in logarithms, Diophantine equation, Thue equation, $abc$-conjecture, primitive divisor, irrationality measure, $p$-adic analysis
The aim of this book is to serve as an introductory text to the theory of linear forms in the logarithms of algebraic numbers, with a special emphasis on a large variety of its applications. We wish to help students and researchers to learn what is hidden inside the blackbox ‚Baker's theory of linear forms in logarithms' (in complex or in $p$-adic logarithms) and how this theory applies to many Diophantine problems, including the e ffective resolution of Diophantine equations, the $abc$-conjecture, and upper bounds for the irrationality measure of some real numbers. Written for a broad audience, this accessible and self-contained book can be used for graduate courses (some 30 exercises are supplied). Specialists will appreciate the inclusion of over 30 open problems and the rich bibliography of over 450 references.
3
14
2018
978-3-03719-183-5
978-3-03719-683-0
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/183
http://www.ems-ph.org/doi/10.4171/183
IRMA Lectures in Mathematics and Theoretical Physics
2523-5133
2523-5141
28
Schubert Varieties, Equivariant Cohomology and Characteristic Classes
Impanga 15
Jarosław
Buczyński
Polish Academy of Sciences and University of Warsaw, Poland
Mateusz
Michałek
Polish Academy of Sciences, Warsaw, Poland and Max Planck-Institute, Leipzig, Germany
Elisa
Postinghel
Loughborough University, UK
Algebraic geometry
Several complex variables and analytic spaces
Algebraic topology
Primary 14-06; secondary 32L10, 14M15, 55N91, 14C17, 14G17
Analytic geometry
IMPANGA, vector bundles, Schubert varieties and degeneracy loci, homogeneous spaces, equivariant cohomology, Thom polynomials, characteristic classes, symmetric functions and polynomials, quasi-elliptic surfaces
IMPANGA stands for the activities of Algebraic Geometers at the Institute of Mathematics, Polish Academy of Sciences, including one of the most important seminars in algebraic geometry in Poland. The topics of the lectures usually fit within the framework of complex algebraic geometry and neighboring areas of mathematics. This volume is a collection of contributions by the participants of the conference IMPANGA15, organized by participants of the seminar, as well as notes from the major lecture series of the seminar in the period 2010–2015. Both original research papers and self-contained expository surveys can be found here. The articles circulate around a broad range of topics within algebraic geometry such as vector bundles, Schubert varieties, degeneracy loci, homogeneous spaces, equivariant cohomology, Thom polynomials, characteristic classes, symmetric functions and polynomials, and algebraic geometry in positive characteristic.
1
21
2018
978-3-03719-182-8
978-3-03719-682-3
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/182
http://www.ems-ph.org/doi/10.4171/182
EMS Series of Congress Reports
2523-515X
2523-5168
Introduction
Jarosław
Buczyński
Polish Academy of Sciences, Warsaw, Poland
Mateusz
Michałek
Polish Academy of Sciences, Warsaw, Poland
Elisa
Postinghel
Loughborough University, UK
1
7
1
10.4171/182-1/1
http://www.ems-ph.org/doi/10.4171/182-1/1
Friedrich Hirzebruch – a handful of reminiscences
Piotr
Pragacz
Polish Academy of Sciences, Warsaw, Poland
9
23
1
10.4171/182-1/2
http://www.ems-ph.org/doi/10.4171/182-1/2
Pieri rule for the factorial Schur $P$-functions
Soojin
Cho
Ajou University, Suwon, Republic of Korea
Takeshi
Ikeda
Okayama University of Science, Japan
Maximal orthogonal Grassmannian, Pieri rule, factorial Schur $P$-functions
Algebraic geometry
Combinatorics
We prove an identity expressing the product of two factorial Schur $P$-functions as an alternating sum of the factorial Schur $P$-functions with explicitly defined coefficients depending on the deformation parameters. As an application, we derive the Pieri rule for the factorial Schur $P$-functions.
25
48
1
10.4171/182-1/3
http://www.ems-ph.org/doi/10.4171/182-1/3
Restriction varieties and the rigidity problem
Izzet
Coskun
University of Illinois at Chicago, USA
Homogeneous varieties, Schubert varieties, quadratic forms, restriction coefficients
Algebraic geometry
Commutative rings and algebras
This is a survey paper based on the author's lectures given at IMPAN in December 2013. We will discuss recent results on the restriction and rigidity problems. The purpose of the lectures was to develop a more geometric approach to the study of isotropic flag varieties. As an illustration of the techniques, we compute the map induced in cohomology of the inclusion of $OG(k,n)$ and $SG(k,n)$ in $G(k,n)$ via an explicit sequence of rational equivalences. We also discuss applications to classifying representatives of Schubert classes.
49
95
1
10.4171/182-1/4
http://www.ems-ph.org/doi/10.4171/182-1/4
On Plücker equations characterizing Grassmann cones
Letterio
Gatto
Politecnico di Torino, Italy
Parham
Salehyan
IBILCE/UNESP, São José do Rio Preto, Brazil
Algebraic geometry
The KP hierarchy (after Kadomtsev and Petshiasvily) is a system of infinitely many PDEs in Lax form defining a universal family of iso-spectral deformation of an ordinary linear differential operator. It is a classical result due to Sato's japanese school that the rational solutions to the KP hierarchy are parametrized by a cone over an infinite-dimensional Grassmann variety. The present survey will revisit this fact from the point of view of Schubert derivations on a Grassmann algebra. These enable to encode the classical Plücker equations of Grassmannians of $r$-dimensional subspaces in a formula whose limit for $r \to \infty$ coincides with the KP hierarchy, phrased in terms of vertex operators, showing in particular how the latter is intimately related to Schubert calculus.
97
125
1
10.4171/182-1/5
http://www.ems-ph.org/doi/10.4171/182-1/5
Kempf–Laksov Schubert classes for even infinitesimal cohomology theories
Thomas
Hudson
Bergische Universität Wuppertal, Germany
Tomoo
Matsumura
Okayama University of Science, Japan
Grassmann bundles, Schubert varieties, Oriented cohomology theories, Chern classes
Algebraic geometry
Combinatorics
n this paper we prove a generalisation of Kempf–Laksov formula for the degeneracy loci classes in the even infinitesimal cohomology theories of Grassmann bundles and Lagrangian Grassmann bundles.
127
151
1
10.4171/182-1/6
http://www.ems-ph.org/doi/10.4171/182-1/6
On the multicanonical systems of quasi-elliptic surfaces in characteristic 3
Toshiyuki
Katsura
Hosei University, Tokyo, Japan
Quasi-elliptic surfaces, multicanonical systems, positive characteristic
Algebraic geometry
We consider the multicanonical systems $\vert mK_S \vert$ of quasi-elliptic surfaces with Kodaira dimension $1$ in characteristic 3. We show that for any $m \geq 5$ $\vert mK_S \vert$ gives the structure of quasi-elliptic fiber space, and the number $5$ is best possible to give the structure for any such surfaces.
153
157
1
10.4171/182-1/7
http://www.ems-ph.org/doi/10.4171/182-1/7
Characteristic classes of mixed Hodge modules and applications
Laurentiu
Maxim
University of Wisconsin, Madison, USA
Jörg
Schürmann
Universität Münster, Germany
Characteristic classes, Atiyah–Singer classes, mixed Hodge modules, $V$-filtration, nearby and vanishing cycles, singularities, toric varieties, hypersurface, symmetric product, generating series.
Manifolds and cell complexes
Algebraic geometry
Associative rings and algebras
Several complex variables and analytic spaces
We give an overview, with an emphasis on applications, of recent developments on the interaction between characteristic class theories for singular spaces and Saito's theory of mixed Hodge modules in the complex algebraic context.
159
202
1
10.4171/182-1/8
http://www.ems-ph.org/doi/10.4171/182-1/8
On a certain family of $U(\mathfrak b)$-modules
Piotr
Pragacz
Polish Academy of Sciences, Warsaw, Poland
$U(\mathfrak b)$-module, Demazure module, KP module, KP filtration, cyclic module, character, Schur function, Schur functor, Schubert polynomial, subquotient, positivity, ample bundle
Algebraic geometry
Group theory and generalizations
We report on results of Kraskiewicz and the author, and Watanabe on KP modules materializing Schubert polynomials, and filtrations having KP modules as their sub quotients. We discuss applications of the bundles $S_w(E)$ for filtered ample bundles $E$ and KP filtrations to positivity due to Fulton and Watanabe respectively.
203
224
1
10.4171/182-1/9
http://www.ems-ph.org/doi/10.4171/182-1/9
Equivariant Chern–Schwartz–MacPherson classes in partial flag varieties: interpolation and formulae
Richárd
Rimányi
University of North Carolina at Chapel Hill, USA
Alexander
Varchenko
University of North Carolina at Chapel Hill, USA
Equivariant Chern–Schwartz–MacPherson class, Schubert calculus, weight function
Algebraic geometry
Nonassociative rings and algebras
Consider the natural torus action on a partial flag manifold $\mathcal F}_\lambda$. Let $\Omega_I\subset \mathcal F}_\lambda$ be an open Schubert variety, and let $c^{sm}(\Omega_I)\in H_T^*(\mathcal F}_\lambda)$ be its torus equivariant Chern–Schwartz–MacPherson class. We show a set of interpolation properties that uniquely determine $c^{sm}(\Omega_I)$, as well as a formula, of 'localization type', for $c^{sm}(\Omega_I)$. In fact, we proved similar results for a class $\kappa_I\in H_T^*(\mathcal F}_\lambda)$ – in the context of quantum group actions on the equivariant cohomology groups of partial flag varieties. In this note we show that $c^{sm}(\Omega_I)=\kappa_I$.
225
235
1
10.4171/182-1/10
http://www.ems-ph.org/doi/10.4171/182-1/10
Thom polynomials in $\mathcal A$-classification I: counting singular projections of a surface
Takahisa
Sasajima
Kyoto, Japan
Toru
Ohmoto
Hokkaido University, Sapporo, Japan
$\mathcal A$-classification of map-germs, Thom polynomials, classical enumerative geometry, projective surfaces
Algebraic geometry
Manifolds and cell complexes
We study universal polynomials of characteristic classes associated to the $\mathcal A$-classification of map-germs $(\mathbb C^2,0) \to (\mathbb C^n, 0)$ $(n=2,3)$, that enable us to systematically generalize enumerative formulae in classical algebraic geometry of projective surfaces in 3 and 4-spaces.
237
259
1
10.4171/182-1/11
http://www.ems-ph.org/doi/10.4171/182-1/11
Schubert polynomials and degeneracy locus formulas
Harry
Tamvakis
University of Maryland, College Park, USA
Schubert polynomials, theta polynomials, symmetric functions, Weyl groups, divided difference operators, flag varieties, degeneracy loci, equivariant cohomology
Algebraic geometry
Combinatorics
In previous work, we employed the approach to Schubert polynomials by Fomin, Stanley, and Kirillov to obtain simple, uniform proofs that the double Schubert polynomials of Lascoux and Schützenberger and Ikeda, Mihalcea, and Naruse represent degeneracy loci for the classical groups in the sense of Fulton. Using this as our starting point, and purely combinatorial methods, we obtain a new proof of the general formulas of [T5], which represent the degeneracy loci coming from any isotropic partial flag variety. Along the way, we also find several new formulas and elucidate the connections between some earlier ones.
261
314
1
10.4171/182-1/12
http://www.ems-ph.org/doi/10.4171/182-1/12
Hirzebruch $\chi_y$-genera of complex algebraic fiber bundles – the multiplicativity of the signature modulo 4
Shoji
Yokura
Kagoshima University, Japan
315
330
1
10.4171/182-1/13
http://www.ems-ph.org/doi/10.4171/182-1/13
Pushing-forward Schur classes using iterated residues at infinity
Magdalena
Zielenkiewicz
University of Warsaw, Poland
Gysin homomorphism, equivariant cohomology, torus action, homogeneous space
Algebraic geometry
Differential geometry
In this paper we review the results presented during the IMPANGA 15 Conference on the author's approach to equivariant Gysin homomorphism via iterated residues at infinity, with connections to the Jeffrey–Kirwan nonabelian localization theorem in symplectic geometry. We show examples of computations using the formulas of our previous paper, which express the push-forwards in equivariant cohomology as iterated residues at infinity. As an example we consider the equivariant cohomology of the complex Lagrangian Grassmannian $LG(n)$ with the action of the maximal torus in the symplectic group $Sp(n)$. In particular, we obtain, via our methods, analogues in equivariant cohomology of some well-known results due to Pragacz and Ratajski on push-forwards of Schur classes on $LG(n)$.
331
345
1
10.4171/182-1/14
http://www.ems-ph.org/doi/10.4171/182-1/14
Regular, Quasi-regular and Induced Representations of Infinite-dimensional Groups
Alexander
Kosyak
National Academy of Science of Ukraine, Kiev, Ukraine
Topological groups, Lie groups
Measure and integration
Probability theory and stochastic processes
22E66, 22E65; 60B15, 28C20
Topology
Quasi-invariant measure on infinite-dimensional group, ergodic measure, Hilbert–Lie group, unitary, irreducible, regular, quasi-regular, induced representations, Ismagilov conjecture, Schur–Weil duality, Kirillov orbit method, von Neumann algebras, factor, type of factors, C*-group algebras, finite field
Almost all harmonic analysis on locally compact groups is based on the existence (and uniqueness) of a Haar measure. Therefore, it is very natural to attempt a similar construction for non-locally compact groups. The essential idea is to replace the non-existing Haar measure on an infinite-dimensional group by a suitable quasi-invariant measure on an appropriate completion of the initial group or on the completion of a homogeneous space. The aim of the book is a systematic development, by example, of noncommutative harmonic analysis on infinite-dimensional (non-locally compact) matrix groups. We generalize the notion of regular, quasi-regular and induced representations for arbitrary infinite-dimensional groups. The central idea to verify the irreducibility is the Ismagilov conjecture. We also extend the Kirillov orbit method for the group of upper triangular matrices of infinite order. In order to make the content accessible to a wide audience of nonspecialists, the exposition is essentially self-contained and very few prerequisites are needed. The book is aimed at graduate and advanced undergraduate students, as well as mathematicians who wish an introduction to representations of infinite-dimensional groups.
5
31
2018
978-3-03719-181-1
978-3-03719-681-6
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/181
http://www.ems-ph.org/doi/10.4171/181
EMS Tracts in Mathematics
Higher-Dimensional Knots According to Michel Kervaire
Françoise
Michel
Université Paul Sabatier, Toulouse, France
Claude
Weber
Université de Genève, Switzerland
Manifolds and cell complexes
Several complex variables and analytic spaces
57Q45, 57R65, 32S55
Algebraic topology
Knots in high dimensions, homotopy spheres, complex singularities
Michel Kervaire wrote six papers which can be considered fundamental to the development of higher-dimensional knot theory. They are not only of historical interest but naturally introduce to some of the essential techniques in this fascinating theory. This book is written to provide graduate students with the basic concepts necessary to read texts in higher-dimensional knot theory and its relations with singularities. The first chapters are devoted to a presentation of Pontrjagin’s construction, surgery and the work of Kervaire and Milnor on homotopy spheres. We pursue with Kervaire’s fundamental work on the group of a knot, knot modules and knot cobordism. We add developments due to Levine. Tools (like open books, handlebodies, plumbings, …) often used but hard to find in original articles are presented in appendices. We conclude with a description of the Kervaire invariant and the consequences of the Hill–Hopkins–Ravenel results in knot theory.
7
25
2017
978-3-03719-180-4
978-3-03719-680-9
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/180
http://www.ems-ph.org/doi/10.4171/180
EMS Series of Lectures in Mathematics
2523-5176
2523-5184
A Course In Error-Correcting Codes
Second edition
Jørn
Justesen
Technical University of Denmark, Lyngby, Denmark
Tom
Høholdt
Technical University of Denmark, Lyngby, Denmark
Information and communication, circuits
Field theory and polynomials
94-01;12-01
Mathematical theory of computation
Fields + rings
Error-correcting codes, Reed–Solomon codes, convolutional codes, product codes, graph codes, algebraic geometry codes
This book, updated and enlarged for the second edition, is written as a text for a course aimed at 3rd or 4th year students. Only some familiarity with elementary linear algebra and probability is directly assumed, but some maturity is required. The students may specialize in discrete mathematics, computer science, or communication engineering. The book is also a suitable introduction to coding theory for researchers from related fields or for professionals who want to supplement their theoretical basis. The book gives the coding basics for working on projects in any of the above areas, but material specific to one of these fields has not been included. The chapters cover the codes and decoding methods that are currently of most interest in research, development, and application. They give a relatively brief presentation of the essential results, emphasizing the interrelations between different methods and proofs of all important results. A sequence of problems at the end of each chapter serves to review the results and give the student an appreciation of the concepts. In addition, some problems and suggestions for projects indicate direction for further work. The presentation encourages the use of programming tools for studying codes, implementing decoding methods, and simulating performance. Specific examples of programming exercises are provided on the book's home page.
7
11
2017
978-3-03719-179-8
978-3-03719-679-3
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/179
http://www.ems-ph.org/doi/10.4171/179
EMS Textbooks in Mathematics
Shape Variation and Optimization
A Geometrical Analysis
Antoine
Henrot
Université de Lorraine, Vandœuvre-lès-Nancy, France
Michel
Pierre
ENS Cachan Bretagne, Bruz, France
Calculus of variations and optimal control; optimization
Partial differential equations
Differential geometry
Global analysis, analysis on manifolds
49Q10, 49Q05, 49Q12, 49K20, 49K40, 53A10, 35R35, 35J20, 58E25, 31B15, 65K10, 93B27, 74P20, 74P15, 74G65, 76M30
Calculus + mathematical analysis
Differential equations
Differential + Riemannian geometry
Shape optimization, optimum design, calculus of variations, variations of domains, Hausdorff convergence, continuity with respect to domains, G-convergence, shape derivative, geometry of optimal shapes, Laplace-Dirichlet problem, Neumann problem, overdetermined problems, isoperimetric inequality, capacity, potential theory, spectral theory, homogenization
Optimizing the shape of an object to make it the most efficient, resistant, streamlined, lightest, noiseless, stealthy or the cheapest is clearly a very old task. But the recent explosion of modeling and scientific computing have given this topic new life. Many new and interesting questions have been asked. A mathematical topic was born – shape optimization (or optimum design). This book provides a self-contained introduction to modern mathematical approaches to shape optimization, relying only on undergraduate level prerequisite but allowing to tackle open questions in this vibrant field. The analytical and geometrical tools and methods for the study of shapes are developed. In particular, the text presents a systematic treatment of shape variations and optimization associated with the Laplace operator and the classical capacity. Emphasis is also put on differentiation with respect to domains and a FAQ on the usual topologies of domains is provided. The book ends with geometrical properties of optimal shapes, including the case where they do not exist.
2
15
2018
978-3-03719-178-1
978-3-03719-678-6
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/178
http://www.ems-ph.org/doi/10.4171/178
EMS Tracts in Mathematics
28
Interviews with the Abel Prize Laureates 2003–2016
Martin
Raussen
Aalborg University, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
01A70, 01A60, 01A61, 01A80, 00A35
History of mathematics
Abel prize, laureates, interviews, history of mathematics, appreciation of mathematics
The Abel Prize was established in 2002 by the Norwegian Ministry of Education and Research. It has been awarded annually to mathematicians in recognition of pioneering scientific achievements. Since the first occasion in 2003, Martin Raussen and Christian Skau have had the opportunity to conduct extensive interviews with the laureates. The interviews were broadcast by Norwegian television; moreover, they have appeared in the membership journals of several mathematical societies. The interviews from the period 2003 – 2016 have now been collected in this edition. They highlight the mathematical achievements of the laureates in a historical perspective and they try to unravel the way in which the world’s most famous mathematicians conceive and judge their results, how they collaborate with peers and students, and how they perceive the importance of mathematics for society.
9
1
2017
978-3-03719-177-4
978-3-03719-677-9
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/177
http://www.ems-ph.org/doi/10.4171/177
Abel Prize 2003: Jean-Pierre Serre
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
1
10
1
10.4171/177-1/1
http://www.ems-ph.org/doi/10.4171/177-1/1
Abel Prize 2004: Sir Michael Francis Atiyah and Isadore M. Singer
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
11
29
1
10.4171/177-1/2
http://www.ems-ph.org/doi/10.4171/177-1/2
Abel Prize 2005: Peter D. Lax
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
31
46
1
10.4171/177-1/3
http://www.ems-ph.org/doi/10.4171/177-1/3
Abel Prize 2006: Lennart Carleson
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
47
60
1
10.4171/177-1/4
http://www.ems-ph.org/doi/10.4171/177-1/4
Abel Prize 2007: Srinivasa S. R. Varadhan
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
61
78
1
10.4171/177-1/5
http://www.ems-ph.org/doi/10.4171/177-1/5
Abel Prize 2008: John Griggs Thompson and Jacques Tits
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
79
96
1
10.4171/177-1/6
http://www.ems-ph.org/doi/10.4171/177-1/6
Abel Prize 2009: Mikhail Gromov
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
97
121
1
10.4171/177-1/7
http://www.ems-ph.org/doi/10.4171/177-1/7
Abel Prize 2010: John Tate
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
123
140
1
10.4171/177-1/8
http://www.ems-ph.org/doi/10.4171/177-1/8
Abel Prize 2011: John Milnor
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
141
159
1
10.4171/177-1/9
http://www.ems-ph.org/doi/10.4171/177-1/9
Abel Prize 2012: Endre Szemerédi
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
161
182
1
10.4171/177-1/10
http://www.ems-ph.org/doi/10.4171/177-1/10
Abel Prize 2013: Pierre Deligne
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
183
200
1
10.4171/177-1/11
http://www.ems-ph.org/doi/10.4171/177-1/11
Abel Prize 2014: Yakov G. Sinai
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
201
217
1
10.4171/177-1/12
http://www.ems-ph.org/doi/10.4171/177-1/12
Abel Prize 2015: John F. Nash, Jr. and Louis Nirenberg
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
219
243
1
10.4171/177-1/13
http://www.ems-ph.org/doi/10.4171/177-1/13
Abel Prize 2016: Sir Andrew J. Wiles
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
245
265
1
10.4171/177-1/14
http://www.ems-ph.org/doi/10.4171/177-1/14
An Imaginary Interview with Niels Henrik Abel
Martin
Raussen
Aalborg Universitet, Denmark
Christian
Skau
Norwegian University of Science and Technology (NTNU), Trondheim, Norway
History and biography
General
267
287
1
10.4171/177-1/15
http://www.ems-ph.org/doi/10.4171/177-1/15
European Congress of Mathematics
Berlin, July 18–22, 2016
Volker
Mehrmann
Technical University Berlin, Germany
Martin
Skutella
Technical University Berlin, Germany
General
00Bxx
Mathematics
Mathematics, European congress of mathematics
The European Congress of Mathematics, held every four years, is a well-established major international mathematical event. Following those in Paris (1992), Budapest (1996), Barcelona (2000), Stockholm (2004), Amsterdam (2008), and Kraków (2012), the Seventh European Congress of Mathematics (7ECM) took place in Berlin, Germany, July 18–22, 2016, with about 1100 participants from all over the world. Ten plenary, thirty-three invited and four special lectures formed the core of the program. As at all the previous EMS congresses, ten outstanding young mathematicians received the EMS prizes in recognition of their research achievements. In addition, two more prizes were awarded: The Felix Klein prize for a remarkable solution of an industrial problem, and – for the second time – the Otto Neugebauer Prize for a highly original and influential piece of work in the history of mathematics. The program was complemented by forty-three minisymposia with about 160 talks as well as contributed talks, spread over all areas of mathematics. Several panel discussions and meetings were organized, covering a variety of issues ranging from the future of mathematical publishing to public awareness of mathematics. These proceedings present extended versions of most of the plenary and invited lectures which were delivered during the congress, providing a permanent record of the best what mathematics offers today.
8
6
2018
978-3-03719-176-7
978-3-03719-676-2
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/176
http://www.ems-ph.org/doi/10.4171/176
How has one, and How could have one approached the diversity of mathematical cultures?
Karine
Chemla
Université Paris Diderot, Paris, France
History and biography
This contribution argues that history of mathematics should take as its object not only knowledge, but also ways of doing mathematics that are collectively shared (what I call “mathematical cultures”), and additionally the connections between the two. I provide evidence showing that there is a history of ways of doing mathematics, and this history suggests that mathematical knowledge takes shape at the same time as practices do. Indeed, ways of doing mathematics do not fall out of the sky. They are shaped and transformed by actors in the process of working out some problems and addressing some issues. They represent one of the outcomes of mathematical research. I further argue that attending to the mathematical culture in the context of which actors worked is essential for interpreting their writings.
1
61
1
10.4171/176-1/1
http://www.ems-ph.org/doi/10.4171/176-1/1
Flexible polyhedra and their volumes
Alexander
Gaifullin
Steklov Mathematical Institute, Moscow, Russia
General
We discuss some recent results on flexible polyhedra and the bellows conjecture, which claims that the volume of any flexible polyhedron is constant during the flexion. Also, we survey main methods and several open problems in this area.
63
83
1
10.4171/176-1/2
http://www.ems-ph.org/doi/10.4171/176-1/2
Boolean Functions: Influence, threshold and noise
Gil
Kalai
Hebrew University, Jerusalem, Israel
General
This lecture studies the analysis of Boolean functions and present a few ideas, results, proofs, and problems. We start with the wider picture of expansion in graphs and then concentrate on the graph of the $n$-dimensional discrete cube $\Omega_n$. Boolean functions are functions from $\Omega_n to {0,1}$. We consider the notion of the influence of variables on Boolean functions. The influence of a variable on a Boolean function is the probability that changing the value of the variable changes the value of the function. We then consider Fourier analysis of real functions on $\Omega_n$ and some applications of Fourier methods. We go on to discuss connections with sharp threshold phenomena, percolation, random graphs, extremal combinatorics, correlation inequalities, and more.
85
110
1
10.4171/176-1/3
http://www.ems-ph.org/doi/10.4171/176-1/3
Quantum Fields and Probability
Antti
Kupiainen
University of Helsinki, Finland
General
111
131
1
10.4171/176-1/4
http://www.ems-ph.org/doi/10.4171/176-1/4
Existence of knotted vortex structures in stationary solutions of the Euler equations
Alberto
Enciso
Consejo Superior de Investigaciones Científicas, Madrid, Spain
Daniel
Peralta-Salas
Consejo Superior de Investigaciones Científicas, Madrid, Spain
General
In this paper, we review recent research on certain geometric aspects of the vortex lines of stationary ideal fluids. We mainly focus on the study of knotted and linked vortex lines and vortex tubes, which is a topic that can be traced back to Lord Kelvin and was popularized by the works of Arnold and Moffatt on topological hydrodynamics in the 1960s. In this context, we provide a leisurely introduction to some recent results concerning the existence of stationary solutions to the Euler equations in Euclidean space with a prescribed set of vortex lines and vortex tubes of arbitrarily complicated topology. The content of this paper overlaps substantially with the one the authors published in the Newsletter of the European Mathematical Society in June 2015.
133
153
1
10.4171/176-1/5
http://www.ems-ph.org/doi/10.4171/176-1/5
Symplectic rigidity and quantum mechanics
Leonid
Polterovich
Tel-Aviv University, Israel
General
I present new links between Symplectic Topology and Quantum Mechanics which have been discovered in the framework of function theory on symplectic manifolds. Recent advances in this emerging theory highlight some rigidity features of the Poisson bracket, a fundamental operation governing the mathematical model of Classical Mechanics. Unexpectedly, the intuition behind this rigidity comes from Quantum Mechanics.
155
179
1
10.4171/176-1/6
http://www.ems-ph.org/doi/10.4171/176-1/6
The topology and geometry of automorphism groups of free groups
Karen
Vogtmann
University of Warwick, Coventry, UK
General
In the 1970s Stallings showed that one could learn a great deal about free groups and their automorphisms by viewing the free groups as fundamental groups of graphs and modeling their automorphisms as homotopy equivalences of graphs. Further impetus for using graphs to study automorphism groups of free groups came from the introduction of a space of graphs, now known as Outer space, on which the group Out$(F_n$)
acts nicely. The study of Outer space and its Out$(F_n$)
action continues to give new information about the structure of Out$(F_n$)
, but has also found surprising connections to many other groups, spaces and seemingly unrelated topics, from phylogenetic trees to cyclic operads and modular forms. In this talk I will highlight various ways these ideas are currently evolving.
181
202
1
10.4171/176-1/7
http://www.ems-ph.org/doi/10.4171/176-1/7
Spectral synthesis in Hilbert spaces of entire functions
Anton
Baranov
St. Petersburg State University, Russia
Yurii
Belov
St. Petersburg State University, Russia
General
We give a survey of recent advances in the theory of spaces of entire functions related to the notion of spectral synthesis. In particular, we discuss a solution of a longstanding problem about spectral synthesis for systems of exponentials in $L^2(-\pi, \pi)$
as well as its generalization to de Branges spaces of entire functions. In the de Branges space setting the problem can be related (via a functional model) to spectral theory of rank one perturbations of compact selfadjoint operators; this leads to unexpected examples of rank one perturbations which do not admit spectral synthesis. Related problems for Fock-type spaces are also considered.
203
218
1
10.4171/176-1/8
http://www.ems-ph.org/doi/10.4171/176-1/8
Periodic waves in unsaturated porous media with hysteresis
Bettina
Detmann
Universität Duisburg-Essen, Germany
Pavel
Krejcí
Academy of Sciences, Prague, Czechia
Elisabetta
Rocca
Università degli Studi di Pavia, Italy
General
We consider a PDE system with degenerate hysteresis describing unsaturated flow in 3D porous media. Assuming that a time periodic forcing is prescribed on the boundary, we prove that a time periodic response exists as long as the amplitude of the forcing terms is small enough to keep the solution within the convexity domain of the hysteresis operator.
219
234
1
10.4171/176-1/9
http://www.ems-ph.org/doi/10.4171/176-1/9
2D Ising model: Correlation functions at criticality via Riemann-type boundary value problems
Dmitry
Chelkak
Ecole Normale Supérieure, Paris, France
General
In this note we overview recent convergence results for correlations in the critical planar nearest-neighbor Ising model. We start with a short discussion of the combinatorics of the model and a definition of fermionic and spinor observables. After that, we illustrate our approach to spin correlations by a derivation of two classical explicit formulae in the infinite-volume limit. Then we describe the convergence results (as the mesh size tends to zero, in arbitrary planar domains) for fermionic correlators [14], energy-density [18] and spin expectations [11]. Finally, we discuss scaling limits of mixed correlators involving spins, disorders and fermions, and the classical fusion rules for them.
235
256
1
10.4171/176-1/10
http://www.ems-ph.org/doi/10.4171/176-1/10
On perverse equivalences and rationality
Joseph
Chuang
City University of London, UK
Radha
Kessar
City University of London, UK
General
We show that perverse equivalences between module categories of finite-dimensional algebras preserve rationality. As an application, we give a connection between some famous conjectures from the modular representation theory of finite groups, namely Broué’s Abelian Defect Group conjecture and Donovan’s Finiteness conjectures.
257
262
1
10.4171/176-1/11
http://www.ems-ph.org/doi/10.4171/176-1/11
Torsion homology growth in arithmetic groups
Nicolas
Bergeron
Université Pierre et Marie Curie, Paris, France
General
Various recent works show that certain arithmetic groups -- that generalize the modular group -- can have `a lot' of torsion in their homology. Among these groups are the finite index (congruence) subgroups of $\mathrm{SL}_3 (\mathbb{Z})$ or $\mathrm{SL}_2 (\mathbb{Z} [i])$. In the latter case homology reduces to abelianization. In particular, for \[\textstyle \Gamma_0 (N) = \left\{ \left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) \in \mathrm{SL}_2 (\mathbb{Z} [i]) \; \big| \; N | c \right\} \quad (N \in \mathbb{Z}[i]), \] one may ask about the structure of the finitely generated $\mathbb{Z}$-module $\Gamma_0 (N) ^{\rm ab} = \Gamma_0 (N) / [\Gamma_0 (N) : \Gamma_0 (N)].$ It has a finite torsion part $\Gamma_0 (N)^{\rm ab}_{\rm tors}$ and Akshay Venkatesh and I have conjectured that, as $N$ tends to $\infty$ among primes, we have: \[ \frac{\log |\Gamma_0 (N)^{\rm ab}_{\rm tors}|}{|N|^2} \to \frac{\lambda }{18\pi}, \quad \lambda=L(2, \chi_{\mathbb{Q} (i)}) = 1- \frac{1}{9} + \frac{1}{25} - \frac{1}{49} + \ldots \] More generally one may ask: \emph{How does the amount of torsion in the homology of an arithmetic group grow with the level $N$?} We propose a conjectural partial answer. This contribution presents ideas for how to attack this conjecture and discusses recent progress towards it. This topic interacts with more classical questions of geometry (analytic torsion, Gromov--Thurston norm, (higher) cost, rank and deficiency gradient \dots) and number theory (BSD conjecture, ABC conjecture~\dots). A big motivation is provided by (one of) Peter Scholze's recent breakthrough(s): \emph{very roughly} a mod $p$ torsion class in $\Gamma_0 (N)^{\rm ab}_{\rm tors}$ parametrizes a field extension $K/\mathbb{Q} (i)$ whose Galois group is a subgroup of $\mathrm{GL}_2 (\overline{\mathbb{F}}_p)$. Moreover, it is anticipated that there is a corresponding 'torsion Langlands program'.
263
287
1
10.4171/176-1/12
http://www.ems-ph.org/doi/10.4171/176-1/12
Positivity and higher Teichmüller theory
Olivier
Guichard
Université de Strasbourg, France
Anna
Wienhard
Universität Heidelberg, Germany and Heidelberg Institute for Theoretical Studies, Heidelberg, Germany
General
We introduce $\Theta$-positivity, a new notion of positivity in real semisimple Lie groups. The notion of $\Theta$-positivity generalizes at the same time Lusztig’s total positivity in split real Lie groups as well as well known concepts of positivity in Lie groups of Hermitian type. We show that there are two other families of Lie groups, SO($p,q$)
for $p \neq q$ and a family of exceptional Lie groups, which admit a $\Theta$-positive structure. We describe key aspects of $\Theta$-positivity and make a connection with representations of surface groups and higher Teichmüller theory.
289
310
1
10.4171/176-1/13
http://www.ems-ph.org/doi/10.4171/176-1/13
Diffusion, optimal transport and Ricci curvature
Giuseppe
Savaré
Università di Pavia, Italy
General
Starting from the pioneering paper of Otto-Villani [73], the link between Optimal Transport and Ricci curvature in smooth Riemannian geometry has been deeply studied [35, 86]. Among the various functional and analytic applications, the point of view of Optimal Transport has played a crucial role in the Lott–Sturm–Villani [69, 84, 85, 87] formulation of a “synthetic” notion of lower Ricci curvature bound, which has been extended from the realm of smooth Riemannian manifold to the general framework of metric measure spaces $(X,\mathrm d,\mathfrak m)$
, i.e., (separable, complete) metric spaces endowed with a finite or locally finite Borel measure \mathfrak m. Lower Ricci curvature bounds can also be captured by the celebrated Bakry–Émery [21] approach based on Markov semigroups, diffusion operators and $\Gamma$-calculus for strongly local Dirichlet forms [22]. We will discuss a series of recent contributions [5, 7–9, 12, 38] showing the link of both the approaches with the metric-variational theory of gradient flows [6] and diffusion equations. As a byproduct, when the Cheeger energy on $(X,\mathrm d,\mathfrak m)$ is quadratic (or, equivalently, the Sobolev space $W^{1,2}(X,\mathrm d,\mathfrak m)$ is Hilbertian), we will show that the two approaches lead to essentially equivalent definitions and to a nice geometric framework suitable for deep analytic results.
311
331
1
10.4171/176-1/14
http://www.ems-ph.org/doi/10.4171/176-1/14
Non-discrete simple locally compact groups
Pierre-Emmanuel
Caprace
Université Catholique de Louvain, Louvain-la-Neuve, Belgium
General
Simple Lie groups and simple algebraic groups over local fields are the most prominent members of the class $\mathcal S$ of compactly generated non-discrete simple locally compact groups. We outline a new trend, which emerged in the past decade, whose purpose is the study of $\mathcal S$ as a whole.
333
354
1
10.4171/176-1/15
http://www.ems-ph.org/doi/10.4171/176-1/15
An invitation to circle actions
Leonor
Godinho
Instituto Superior Técnico, Lisboa, Portugal
Silvia
Sabatini
Universität Köln, Germany
General
The problem of determining whether a manifold admits symmetries has been widely studied in mathematics and physics. It is in general hard to determine whether, given a Lie group $G$ and a manifold $M$, there exists a nontrivial action of $G$ on $M$ that preserves a prescribed structure. When $M$ is symplectic, for instance when $M$ is the phase space of a particle, having one conserved quantity whose associated (Hamiltonian) flow on the manifold is periodic, is equivalent to having a Hamiltonian circle action. The following questions are therefore natural: Which symplectic manifolds admit symplectic circle actions? What are their topological properties? Here we discuss these problems when the fixed point set is discrete. In particular we review some of the consequences of the fact that the Chern number $c_1c_{n-1}[M]$ is completely determined by the fixed point data of the action. For example, this allows us to construct an algorithm which, in some cases where the fixed point set is minimal, determines the possible representations at the fixed points of Hamiltonian circle actions with discrete fixed point sets. In general it can be applied towards proving the symplectic Petrie conjecture on these actions. Another application is the obtainment of lower bounds and divisibility results for the number of fixed points on the broader category of almost complex manifolds. This lower bound problem is related to the Kosniowski conjecture, which has been open since 1979. Finally, we show how to extend the 12 and 24 Theorems on the number of lattice points of reflexive polytopes of dimensions 2 and 3 to Delzant reflexive polytopes of any dimension.
355
371
1
10.4171/176-1/16
http://www.ems-ph.org/doi/10.4171/176-1/16
Sampling and interpolating sequences in finite dimensional spaces
Joaquim
Ortega-Cerdà
Universitat de Barcelona, Spain
General
We illustrate how the use of techniques from optimal transport on the study of sampling and interpolation families in finite dimensional spaces provide alternative descriptions of these sequences.
373
387
1
10.4171/176-1/17
http://www.ems-ph.org/doi/10.4171/176-1/17
Weak hyperbolic structures and robust properties of diffeomorphisms and flows
Christian
Bonatti
Université de Bourgogne, Dijon, France
Adriana
da Luz
Universidad de la República, Montevideo, Uruguay, and Université de Bourgogne, Dijon, France
General
A property of a dynamical system is called $C^r$-robust if it holds on a $C^r$-open set of systems. For diffeomorphisms or for non-singular flows, there are many results relating $C^1$-robust properties and global structures of the dynamics, as hyperbolicity, partial hyperbolicity, dominated splitting. However, a difficulty appears when a robust property of a flow holds on a set containing recurrent orbits accumulating a singular point. This phenomenon is now mainly understood in dimension 3, but is still wide open in higher dimensions. Here, we propose a general procedure for adapting the usual hyperbolic structures to the singularities, opening the door for bypassing the difficulty of the coexistence of singular and regular orbits. In particular, this new definition allows us to adapt the proof in [15] to get a characterization of star flows on a $C^1$-open and dense set.
389
405
1
10.4171/176-1/18
http://www.ems-ph.org/doi/10.4171/176-1/18
Free group rings and derived functors
Roman
Mikhailov
Steklov Mathematical Institute, St. Petersburg, Russia
Inder Bir
Passi
Panjab University, Chandigarh, India
General
An approach to identify the normal subgroups determined by ideals in free group rings with the help of the derived functors of non-additive functors is explored. A similar approach, i.e., via derived functors, for computing limits of functors from the category of free presentations to the category of abelian groups, arising from commutator structure of free groups, is also discussed.
407
425
1
10.4171/176-1/19
http://www.ems-ph.org/doi/10.4171/176-1/19
Congested transport at microscopic and macroscopic scales
Bertrand
Maury
Université Paris-Sud, Orsay, France
General
This note addresses mathematical issues raised by congestion constraints in transport equations that arise in the modeling of crowd motion or more general active entities. We address in particular the differences between the microscopic and the macroscopic settings.
427
442
1
10.4171/176-1/20
http://www.ems-ph.org/doi/10.4171/176-1/20
Complex Brunn–Minkowski inequalities and their applications in geometry
Bo
Berndtsson
Chalmers University of Technology, Göteborg, Sweden
General
We survey some positivity results for direct images of line bundles, emphasising their similarities with the classical theorems of Brunn–Minkowski and Prékopa. We also give examples how these results can be applied in complex geometry.
443
457
1
10.4171/176-1/21
http://www.ems-ph.org/doi/10.4171/176-1/21
Counting Steiner Triple Systems
Peter
Keevash
University of Oxford, UK
General
We prove a conjecture of Wilson from 1974 on the number of Steiner Triple Systems. The proof illustrates our method of Randomised Algebraic Construction, which we developed recently to resolve a question of Steiner from 1853 on the existence of combinatorial designs.
459
481
1
10.4171/176-1/22
http://www.ems-ph.org/doi/10.4171/176-1/22
Amenability versus non amenability: An introduction to von Neumann algebras
Stefaan
Vaes
Katholieke Universiteit Leuven, Belgium
General
The theory of von Neumann algebras was initiated by Murray and von Neumann and has deep connections to several areas of mathematics, in particular group theory and ergodic theory. Amenable von Neumann algebras were completely classified by Connes and Haagerup, while numerous classification theorems in the non amenable case were obtained within Popa’s deformation/rigidity theory. This survey article provides an introduction to von Neumann algebras, written for non specialists and with the dichotomy between amenability and non amenability as our guide.
483
500
1
10.4171/176-1/23
http://www.ems-ph.org/doi/10.4171/176-1/23
Recent progress in nonlinear potential theory
Giuseppe
Mingione
Università di Parma, Italy
General
Nonlinear Potential Theory aims at reproducing, in the nonlinear setting, the classical results of potential theory concerning the fine and regularity properties of solutions to linear elliptic and parabolic equations. Potential estimates, integrability, differentiability and continuity properties of solutions are at the heart of the matter. Here we give a brief survey of a few recent results.
501
524
1
10.4171/176-1/24
http://www.ems-ph.org/doi/10.4171/176-1/24
Stochastic dynamics for adaptation and evolution of microorganisms
Sylvain
Billiard
Université des Sciences et Technologies de Lille, Villeneuve-d'Ascq, France
Pierre
Collet
École Polytechnique, Palaiseau, France
Régis
Ferrière
École Normale Supérieure, Paris, France
Sylvie
Méléard
Ecole Polytechnique, Palaiseau, France
Viet Chi
Tran
Université des Sciences et Technologies de Lille, Villeneuve-d'Ascq, France
General
We present a model for the dynamics of a population of bacteria with a continuum of traits, who compete for resources and exchange horizontally (transfer) an otherwise vertically inherited trait with possible mutations. Competition influences individual demographics, affecting population size, which feeds back on the dynamics of transfer. We consider a stochastic individual-based pure jump process taking values in the space of point measures, and whose jump events describe the individual reproduction, transfer and death mechanisms. In a large population scale, the stochastic process is proved to converge to the solution of a nonlinear integro-differential equation. When there are only two different traits and no mutation, this equation reduces to a non-standard two-dimensional dynamical system. We show how crucial the forms of the transfer rates are for the long-term behavior of its solutions. We describe the dynamics of invasion and fixation when one of the two traits is initially rare, and compute the invasion probabilities. Then, we study the process under the assumption of rare mutations. We prove that the stochastic process at the mutation time scale converges to a jump process which describes the successive invasions of successful mutants. We show that the horizontal transfer can have a major impact on the distribution of the successive mutational fixations, leading to dramatically different behaviors, from expected evolution scenarios to evolutionary suicide. Simulations are given to illustrate these phenomena.
525
550
1
10.4171/176-1/25
http://www.ems-ph.org/doi/10.4171/176-1/25
Learning and sparse control of multiagent systems
Massimo
Fornasier
TU München, Garching, Germany
General
In the past decade there has been a large scope of studies on mathematical models of social dynamics. Self-organization, i.e., the autonomous formation of patterns, has been so far the main driving concept. Usually first or second order models are considered with given predetermined nonlocal interaction potentials, tuned to reproduce, at least qualitatively, certain global patterns (such as flocks of birds, milling school of fish or line formations in pedestrian flows etc.). However, it is common experience that self-organization of a society does not always spontaneously occur. In the first part of this survey paper we address the question of whether it is possible to externally and parsimoniously influence the dynamics, to promote the formation of certain desired patterns. In particular we address the issue of finding the sparsest control strategy for finite agent models in order to lead the dynamics optimally towards a given outcome. In the second part of the paper we show the rigorous limit process connecting finite dimensional sparse optimal control problems with ODE constraints to an infinite dimensional sparse mean-field optimal control problem with a constraint given by a PDE of Vlasov-type, governing the dynamics of the probability distribution of the agent population. Moreover, often in practice we do not dispose of a precise knowledge of the governing dynamics. In the last part of this paper we present a variational and optimal transport framework leading to an algorithmic solution to the problem of learning the interaction potentials from the observation of the dynamics of a multiagent system.
551
581
1
10.4171/176-1/26
http://www.ems-ph.org/doi/10.4171/176-1/26
Variational modeling of dislocations in crystals in the line-tension limit
Pilar
Ariza
Universidad de Sevilla, Spain
Sergio
Conti
Universität Bonn, Germany
Adriana
Garroni
Università di Roma La Sapienza, Italy
Michael
Ortiz
California Institute of Technology, Pasadena, USA
General
Dislocations are line singularities in crystals which are crucial for the plastic deformation of materials. Mathematically they can be modeled as measures supported on rectifiable curves, or as vector-valued one-currents. They have a lattice-valued multiplicity, which is a conserved quantity, in the sense that the divergence of the measure (or the boundary of the current) vanishes. Dislocations are necessarily accompanied by large elastic strains, and indeed their energetics can be understood starting from the theory of elasticity, in an appropriate scaling regime. We discuss recent progress in the rigorous derivation of dislocation models in the line-tension regime from linear elasticity, and their application to specific problems in metals. We present numerical simulations on dislocations in bcc molybdenum which show how our line-tension model provides a simple and efficient description of dislocation structures.
583
598
1
10.4171/176-1/27
http://www.ems-ph.org/doi/10.4171/176-1/27
Phase transitions in discrete structures
Amin
Coja-Oghlan
J.W. Goethe-Universität, Frankfurt a.M., Germany
General
Many important parameters of random discrete structures such as random graphs, formulas and codes undergo phase transitions. While some phase transitions such as the emergence of a giant component in the Erd˝os-Rényi random graph are very well understood, other, less understood ones resemble phase transitions in statistical physics models of disordered systems. This paper gives an impression of what we know about this latter class with the particular example of the random graph colouring problem.
599
618
1
10.4171/176-1/28
http://www.ems-ph.org/doi/10.4171/176-1/28
Transverse stability issues in Hamiltonian PDE
Nikolay
Tzvetkov
Université de Cergy-Pontoise, France
General
We present results concerning the transverse stability of one dimensional solitary waves subject to periodic transverse perturbations in the context of the KP equations and the water-waves system.
601
639
1
10.4171/176-1/29
http://www.ems-ph.org/doi/10.4171/176-1/29
Digits of primes
James
Maynard
University of Oxford, UK
General
We discuss some different results on the digits of prime numbers, giving a simplified proof of weak forms of a result of Maynard and Mauduit-Rivat.
641
661
1
10.4171/176-1/30
http://www.ems-ph.org/doi/10.4171/176-1/30
The Hodge theory of the Hecke category
Geordie
Williamson
University of Sydney, Australia
General
Ideas from Hodge theory have found important applications in representation theory. We give a survey of joint work with Ben Elias which uncovers Hodge theoretic structure in the Hecke category (“Soergel bimodules”). We also outline similarities and differences to other combinatorial Hodge theories.
663
683
1
10.4171/176-1/31
http://www.ems-ph.org/doi/10.4171/176-1/31
Additive combinatorics and graph theory
Endre
Szemerédi
Hungarian Academy of Sciences, Budapest, Hungary
General
In this survey we mainly discuss the conditions for a set to contain long arithmetic progressions. We mention the ergodic, Fourier analytic and combinatorial approach to provide such conditions. In addition we describe defferent graph theoretic results with various applications in additive combinatorics and computer science. Finally, we mention some elementary results in discrete geometry.
685
716
1
10.4171/176-1/32
http://www.ems-ph.org/doi/10.4171/176-1/32
The arithmetic and topology of differential equations
Don
Zagier
Max-Planck-Institut für Mathematik, Bonn, Germany
General
This survey paper attempts to present in as elementary a way as possible a wide panorama of results concerning the relations between differential equations on the one hand and algebraic geometry, number theory and topology on the other. We use the famous Apéry numbers as a running example to illustrate the connections with, among other things, the theory of periods (Picard–Fuchs differential equations), the theory of modular forms (and the special values of their L-series), the theory of motives (starting with counting points on varieties over finite fields), and mirror symmetry (in particular, the Gamma Conjecture relating the asymptotics of solutions of differential equations to the multiplicative “Gamma class” of a variety). A number of relations to works by Friedrich Hirzebruch, in whose honor the lecture was given, are also described.
717
776
1
10.4171/176-1/33
http://www.ems-ph.org/doi/10.4171/176-1/33
Two-scale space-time methods for computational solid mechanics
Patrice
Hauret
Michelin Centre de Technologies de Ladoux, Clermont-Ferrand, France
Eric
Lignon
Michelin Centre de Technologies de Ladoux, Clermont-Ferrand, France
Nicole
Spillane
Ecole Polytechnique, Palaiseau, France
Benoît
Pouliot
Université Laval, Québec, Canada
General
The efficient, robust and accurate assessment of structures in large deformation simultaneously requires: i) the resolution of micro-scale states to avoid the use of empirical material laws and assess reliability, ii) the availability of sufficiently light models to enable optimal structure design and uncertainty quantification. The present work contributes to the first objective by the use of variational integrators, a non-conforming space discretization in the sense of mortar methods and the design of optimal coarse grids to enhance traditional domain decomposition methods. The second issue is handled by an homogenized problem iteratively improved by accurate subgrid models in space and time. Several aspects of the method are analyzed and some examples are displayed as an illustration.
777
794
1
10.4171/176-1/34
http://www.ems-ph.org/doi/10.4171/176-1/34
Living mathematics: Poincaré and Weyl in context
Jeremy
Gray
The Open University, Milton Keynes, UK
General
Henri Poincaré and Hermann Weyl enriched both mathematics and physics. Indeed, Poincaré and Weyl lived their mathematics, physics, and philosophy, and they reflected deeply on their work in their popular essays. By looking at their popular writings we can gain an intimate sense of what animated them, the different sets of values and aspirations that they had, and the ways they saw the significance of their work. Tradition is for the mathematician to create, change, even transcend – and surely Poincaré and Weyl transcended it – and for the historian to take apart, complicate, re-balance, even reject. But everyone is marked by their time and place.
795
812
1
10.4171/176-1/35
http://www.ems-ph.org/doi/10.4171/176-1/35
Mesoscopic models in biology
Vincent
Calvez
Université Claude Bernard Lyon 1, Villeurbanne, France
General
I review some mathematical problems which arose in the field of biological sciences, including microbiology, population dynamics and evolutionary biology. My focus is on partial differential equation (PDE) models with multiple levels of description. For each case study, I mean to describe the biological essence of the problem, and the mathematical questions associated with it. I also discuss how mathematical answers may enhance biological knowledge.
813
831
1
10.4171/176-1/36
http://www.ems-ph.org/doi/10.4171/176-1/36
On the singular part of measures constrained by linear PDEs and applications
Guido
De Philippis
SISSA, Trieste, Italy
General
The aim of this note is to present some recent results on the structure of the singular part of measures satisfying a PDE constraint, and to describe some applications.
833
845
1
10.4171/176-1/37
http://www.ems-ph.org/doi/10.4171/176-1/37
Recent progress on Bernoulli convolutions
Péter
Varjú
University of Cambridge, UK
General
The Bernoulli convolution with parameter $\lambda \in (0,1)$ is the measure on $\mathbf R$ that is the distribution of the random power series $\Sigma ± \lambda^n$, where ± are independent fair coin-tosses. This paper surveys recent progress on our understanding of the regularity properties of these measures.
847
867
1
10.4171/176-1/38
http://www.ems-ph.org/doi/10.4171/176-1/38
Random currents expansion of the Ising model
Hugo
Duminil-Copin
Institut des Hautes Études Scientifiques (IHÉS), Bures-sur-Yvette, France and Université de Genève, Switzerland
General
Critical behavior at an order/disorder phase transition has been a central object of interest in statistical physics. In the past century, techniques borrowed from many different fields of mathematics (Algebra, Combinatorics, Probability, Complex Analysis, Spectral Theory, etc) have contributed to a more and more elaborate description of the possible critical behaviors for a large variety of models. The Ising model is maybe one of the most striking success of this cross-fertilization, for this model of ferromagnetism is now very well understood both physically and mathematically. In this article, we review an approach, initiated in [7, 24] and based on the notion of random currents, enabling a deep study of the model.
869
889
1
10.4171/176-1/39
http://www.ems-ph.org/doi/10.4171/176-1/39
Functional Analysis and Operator Theory for Quantum Physics
The Pavel Exner Anniversary Volume
Jaroslav
Dittrich
Czech Academy of Sciences, Rez-Prague, Czech Republic
Hynek
Kovařík
Università degli Studi di Brescia, Italy
Ari
Laptev
Imperial College London, UK
Quantum theory
Partial differential equations
81Q37, 81Q35, 35P15, 35P25
Quantum physics (quantum mechanics)
Differential equations
Schrödinger operators, point interactions, metric graphs, quantum waveguides, eigenvalue estimates, operator-valued functions, Cayley–Hamilton theorem, adiabatic theorem
This volume is dedicated to Pavel Exner on the occasion of his 70th anniversary. It collects contributions by numerous scientists with expertise in mathematical physics and in particular in problems arising from quantum mechanics. The questions addressed in the contributions cover a large range of topics. A lot of attention was paid to differential operators with zero range interactions, which are often used as models in quantum mechanics. Several authors considered problems related to systems with mixed-dimensions such as quantum waveguides, quantum layers and quantum graphs. Eigenvalues and eigenfunctions of Laplace and Schrödinger operators are discussed too, as well as systems with adiabatic time evolution. Although most of the problems treated in the book have a quantum mechanical background, some contributions deal with issues which go well beyond this framework; for example the Cayley–Hamilton theorem, approximation formulae for contraction semigroups or factorization of analytic operator-valued Fredholm functions. As for the mathematical tools involved, the book provides a wide variety of techniques from functional analysis and operator theory. Altogether the volume presents a collection of research papers which will be of interest to any active scientist working in one of the above mentioned fields.
5
5
2017
978-3-03719-175-0
978-3-03719-675-5
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/175
http://www.ems-ph.org/doi/10.4171/175
EMS Series of Congress Reports
2523-515X
2523-5168
Relative partition function of Coulomb plus delta interaction
Sergio
Albeverio
Universität Bonn, Germany
Claudio
Cacciapuoti
Università dell'Insubria, Como, Italy
Mauro
Spreafico
Università del Salento & INFN, Lecce, Italy
Relative zeta function, relative partition function, relative spectral measures, Coulomb interaction, point interactions, zeta regularization, finite temperature quantum fields, Casimir effect, asymptotic expansions
Global analysis, analysis on manifolds
Number theory
Quantum theory
The relative partition function and the relative zeta function of the perturbation of the Laplace operator by a Coulomb potential plus a point interaction centered in the origin is discussed. Applications to the study of the Casimir effect are indicated.
1
29
1
10.4171/175-1/1
http://www.ems-ph.org/doi/10.4171/175-1/1
Inequivalence of quantum Dirac fields of different masses and the underlying general structures involved
Asao
Arai
Hokkaido University, Sapporo, Japan
Canonical anticommutation relations, fermion Fock space, inequivalent representation, mass, quantum Dirac field
Quantum theory
Operator theory
A family of irreducible representations of the canonical anticommutation relations over an abstract Hilbert space indexed by a set of bounded linear operators is presented and a theorem on the mutual equivalence of them is established. As an application of the theorem, it is proved that quantum Dirac fields of different masses are mutually inequivalent. Moreover, a new class of irreducible representations of the CAR over a Hilbert space, which includes, as a special case, time-zero quantum Dirac fields, is constructed.
31
53
1
10.4171/175-1/2
http://www.ems-ph.org/doi/10.4171/175-1/2
On a class of Schrödinger operators exhibiting spectral transition
Diana
Barseghyan
University of Ostrava, Czech Republic
Olga
Rossi
University of Ostrava, Czech Republic
Spectral transition, discrete spectrum, eigenvalue estimates
Quantum theory
Partial differential equations
We show that the operator for $\lambda\frac{\pi^2}{4}$ its spectrum contains the real line. In the critical case $\lambda=\frac{\pi^2}{4}$ we prove that the spectrum coincides with the half line $[0, \infty)$.
55
70
1
10.4171/175-1/3
http://www.ems-ph.org/doi/10.4171/175-1/3
On the quantum mechanical three-body problem with zero-range interactions
Giulia
Basti
Università di Roma La Sapienza, Italy
Alessandro
Teta
Università di Roma La Sapienza, Italy
Point interactions, self-adjoint extensions, unitary gas, Thomas effect
Quantum theory
Functional analysis
Operator theory
In this note we discuss the quantum mechanical three-body problem with pairwise zero-range interactions in dimension three. We review the state of the art concerning the construction of the corresponding Hamiltonian as a self-adjoint operator in the bosonic and in the fermionic case. Exploiting a quadratic form method, we also prove self-adjointness and boundedness from below in the case of three identical bosons when the Hilbert space is suitably restricted, i.e., excluding the "s-wave“ subspace.
71
93
1
10.4171/175-1/4
http://www.ems-ph.org/doi/10.4171/175-1/4
On the index of meromorphic operator-valued functions and some applications
Jussi
Behrndt
TU Graz, Austria
Fritz
Gesztesy
Baylor University, Waco, USA
Helge
Holden
University of Trondheim, Norway
Roger
Nichols
The University of Tennessee at Chattanooga, USA
Factorization of operator-valued analytic functions, multiplicity of eigenvalues, index computations for finitely meromorphic operator-valued functions, Birman–Schwinger operators, dual pairs
Operator theory
We revisit and connect several notions of algebraic multiplicities of zeros of analytic operator-valued functions and discuss the concept of the index of meromorphic operator-valued functions in complex, separable Hilbert spaces. Applications to abstract perturbation theory and associated Birman–Schwinger-type operators and to the operator-valued Weyl–Titchmarsh functions associated to closed extensions of dual pairs of closed operators are provided.
95
127
1
10.4171/175-1/5
http://www.ems-ph.org/doi/10.4171/175-1/5
Trace formulae for Schrödinger operators with singular interactions
Jussi
Behrndt
TU Graz, Austria
Matthias
Langer
University of Strathclyde, Glasgow, UK
Vladimir
Lotoreichik
Nuclear Physics Institute, Řež - Prague, Czech Republic
Trace formula, delta interaction, Schrödinger operator, singular potential
Partial differential equations
Quantum theory
Operator theory
Let $\Sigma\subset\mathbb R^d$ be a $C^\infty$-smooth closed compact hypersurface, which splits the Euclidean space $\mathbb R^d$ into two domains $\Omega_\pm$. In this note self-adjoint Schrödinger operators with $\delta$ and $\delta'$-interactions supported on $\Sigma$ are studied. For large enough $m\in\mathbb N$ the difference of $m$th powers of resolvents of such a Schrödinger operator and the free Laplacian is known to belong to the trace class. We prove trace formulae, in which the trace of the resolvent power difference in $L^2(\mathbb R^d)$ is written in terms of Neumann-to-Dirichlet maps on the boundary space $L^2(\Sigma)$.
129
152
1
10.4171/175-1/6
http://www.ems-ph.org/doi/10.4171/175-1/6
An improved bound for the non-existence of radial solutions of the Brezis–Nirenberg problem in $\mathbb H^n$
Rafael
Benguria
Pontificia Universidad Católica de Chile, Santiago de Chile, Chile
Soledad
Benguria
University of Wisconsin, Madison, USA
Brezis–Nirenberg problem, hyperbolic space, nonexistence of solutions, Pohozaev identity, Hardy inequality
Partial differential equations
Using a Rellich–Pohozaev argument and Hardy's inequality, we derive an improved bound on the nonlinear eigenvalue for the non existence of radial solutions of a Brezis–Nirenberg problem, with Dirichlet boundary conditions, on a geodesic ball of $\mathbb{H}^n$, for $2
153
160
1
10.4171/175-1/7
http://www.ems-ph.org/doi/10.4171/175-1/7
Twisted waveguide with a Neumann window
Philippe
Briet
Université de Toulon, La Garde, France
Hiba
Hammedi
Université de Toulon et du Var, La Garde, France
Waveguide, mixed boundary conditions, twisting
Quantum theory
Operator theory
This paper is concerned with the study of the existence/non-existence of the discrete spectrum of the Laplace operator on a domain of $\mathbb R ^3$ which consists in a twisted tube. This operator is defined by means of mixed boundary conditions. Here we impose Neumann Boundary conditions on a bounded open subset of the boundary of the domain (the Neumann window) and Dirichlet boundary conditions elsewhere.
161
175
1
10.4171/175-1/8
http://www.ems-ph.org/doi/10.4171/175-1/8
Example of a periodic Neumann waveguide with a gap in its spectrum
Giuseppe
Cardone
Università del Sannio, Benevento, Italy
Andrii
Khrabustovskyi
Karlsruher Institut für Technologie, Germany
Periodic waveguides, spectral gaps, asymptotic analysis
Partial differential equations
Operator theory
In this note we investigate spectral properties of a periodic waveguide $\Omega^\varepsilon$ ($\varepsilon$ is a small parameter) obtained from a straight strip by attaching an array of $\varepsilon$-periodically distributed identical protuberances having "room-and-passage" geometry. In the current work we consider the operator $\mathcal{A}^\varepsilon =-\rho^\varepsilon\Delta_{\Omega^\varepsilon}$, where $\Delta_{\Omega^\varepsilon}$ is the Neumann Laplacian in $\Omega^\varepsilon$, the weight $\rho^\varepsilon$ is equal to $1$ everywhere except the union of the „rooms". We will prove that the spectrum of $\mathcal{A}^\varepsilon$ has at least one gap as $\varepsilon$ is small enough provided certain conditions on the weight $\rho^\varepsilon$ and the sizes of attached protuberances hold.
177
187
1
10.4171/175-1/9
http://www.ems-ph.org/doi/10.4171/175-1/9
Two-dimensional time-dependent point interactions
Raffaele
Carlone
Università degli Studi di Napoli Federico II, Italy
Michele
Correggi
Università degli Studi Roma Tre, Italy
Rodolfo
Figari
Università degli Studi di Napoli Federico II, Italy
Partial differential equations
Quantum theory
We study the time-evolution of a quantum particle subjected to time-dependent zero-range forces in two dimensions. After establishing a conceivable ansatz for the solution to the Schrödinger equation, we prove that the wave packet time-evolution is completely specified by the solutions of a system of Volterra-type equations – the charge equations – involving the coefficients of the singular part of the wave function, thus extending to the two-dimensional case known results in one and three dimensions.
189
211
1
10.4171/175-1/10
http://www.ems-ph.org/doi/10.4171/175-1/10
On resonant spectral gaps in quantum graphs
Ngoc
Do
Texas A&M University, College Station, USA
Peter
Kuchment
Texas A&M University, College Station, USA
Beng
Ong
CGG, Houston, USA
Quantum graph, spectral gap, resonator
Quantum theory
Partial differential equations
Global analysis, analysis on manifolds
In this brief paper we present some results on creating and manipulating spectral gaps for a (regular) quantum graph by inserting appropriate internal structures into its vertices. Complete proofs and extensions of the results are planned for another publication.
213
222
1
10.4171/175-1/11
http://www.ems-ph.org/doi/10.4171/175-1/11
Adiabatic theorem for a class of stochastic differential equations on a Hilbert space
Martin
Fraas
Albany, USA
Partial differential equations
Quantum theory
We derive an adiabatic theory for a stochastic differential equation, $$ \varepsilon\, \d X(s) = L_1(s) X(s)\, \d s + \sqrt{\varepsilon} L_2(s) X(s) \, \d B_s, $$ under a condition that instantaneous stationary states of $L_1(s)$ are also stationary states of $L_2(s)$. We use our results to derive the full statistics of tunneling for a driven stochastic Schrödinger equation describing a dephasing process.
223
243
1
10.4171/175-1/12
http://www.ems-ph.org/doi/10.4171/175-1/12
Eigenvalues of Schrödinger operators with complex surface potentials
Rupert
Frank
Caltech, Pasadena, United States
Partial differential equations
Quantum theory
We consider Schrödinger operators in $\mathbb R^d$ with complex potentials supported on a hyperplane and show that all eigenvalues lie in a disk in the complex plane with radius bounded in terms of the $L^p$ norm of the potential with $d-1 < p \leq d$. We also prove bounds on sums of powers of eigenvalues.
245
259
1
10.4171/175-1/13
http://www.ems-ph.org/doi/10.4171/175-1/13
A lower bound to the spectral threshold in curved quantum layers
Pedro
Freitas
Universidade de Lisboa, Portugal
David
Krejčiřík
Czech Technical University in Prague, Prague, Czech Republic
Dirichlet Laplacian in tubes, quantum waveguides, quantum layers, ground-state energy
Number theory
Algebraic geometry
We derive a lower bound to the spectral threshold of the Dirichlet Laplacian in tubular neighbourhoods of constant radius about complete surfaces. This lower bound is given by the lowest eigenvalue of a one-dimensional operator depending on the radius and principal curvatures of the reference surface. Moreover, we show that it is optimal if the reference surface is non-negatively curved.
261
269
1
10.4171/175-1/14
http://www.ems-ph.org/doi/10.4171/175-1/14
To the spectral theory of vector-valued Sturm–Liouville operators with summable potentials and point interactions
Yaroslav
Granovskyi
National Academy of Science of Ukraine, Slavyansk, Ukraine
Mark
Malamud
National Academy of Science of Ukraine, Slavyansk, Ukraine
Hagen
Neidhardt
Karl-Weierstraß-Institut für Mathematik, Berlin, Germany
Andrea
Posilicano
Università dell'Insubria, Como, Italy
Vector-valued Sturm–Liouville operator, point interaction, spectrum, absolutely and singular continuous spectrum, eigenvalues, boundary triplets, Weyl function
Ordinary differential equations
Operator theory
The paper is devoted to the spectral theory of vector-valued Sturm–Liouville operators on the half-line with a summable potential and a finite number of point interactions. It is shown that the positive spectrum is purely absolutely continuous and of constant multiplicity. The negative spectrum is either finite or discrete with the only accumulation point at zero. Our approach relies on the thorough investigation of the corresponding Weyl functions and involves technique elaborated in our previous papers.
271
313
1
10.4171/175-1/15
http://www.ems-ph.org/doi/10.4171/175-1/15
Spectral asymptotics for the Dirichlet Laplacian with a Neumann window via a Birman–Schwinger analysis of the Dirichlet-to-Neumann operator
André
Hänel
Leibniz Universität Hannover, Germany
Timo
Weidl
Universität Stuttgart, Germany
Laplacian, Neumann window, spectral asymptotics, Dirichlet-to-Neumann operator, Birman–Schwinger analysis
Partial differential equations
Quantum theory
In the present article we will give a new proof of the ground state asymptotics of the Dirichlet Laplacian with a Neumann window acting on functions which are defined on a two-dimensional infinite strip or a three-dimensional infinite layer. The proof is based on the analysis of the corresponding Dirichlet-to-Neumann operator as a first order classical pseudo-differential operator. Using the explicit representation of its symbol we prove an asymptotic expansion as the window length decreases.
315
352
1
10.4171/175-1/16
http://www.ems-ph.org/doi/10.4171/175-1/16
Dirichlet eigenfunctions in the cube, sharpening the Courant nodal inequality
Bernard
Helffer
Université de Nantes, France
Rola
Kiwan
American University in Dubai, United Arab Emirates
Courant theorem, nodal lines, nodal domains, Dirichlet, cube
Partial differential equations
Global analysis, analysis on manifolds
This paper is devoted to the refined analysis of Courant's theorem for the Dirichlet Laplacian in a bounded open set. Starting from the work by A. Pleijel in 1956, many papers have investigated in which cases the inequality in Courant's theorem is an equality. All these results were established for open sets in $\mathbb R^2$ or for surfaces like $\mathbb S^2$ or $\mathbb T^2$. The aim of the current paper is to look for the case of the cube in $\mathbb R^3$. We will prove that the only eigenvalues of the Dirichlet Laplacian which are Courant sharp are the two first eigenvalues.
353
371
1
10.4171/175-1/17
http://www.ems-ph.org/doi/10.4171/175-1/17
A mathematical modeling of electron–phonon interaction for small wave numbers close to zero
Masao
Hirokawa
Hiroshima University, Japan
Electron-phonon interaction, Carleman operator, infrared catastrophe
Quantum theory
This paper is dedicated to Pavel Exner on the occasion of his 70th birthday by introducing a part of our attempts in material science, that is, a mathematical modeling of electron-phonon interaction for small wave numbers close to zero. We extrapolate the electron-phonon interaction on the presupposition that the phonon dispersion relation and the electron-phonon coupling function are estimated using some experimental data.
373
399
1
10.4171/175-1/18
http://www.ems-ph.org/doi/10.4171/175-1/18
The modified unitary Trotter–Kato and Zeno product formulas revisited
Takashi
Ichinose
Kanazawa University, Japan
Trotter product formula, Trotter–Kato product formula, Zeno product formula, unitary groups, resolvents, form sum of selfadjoint operators, Feynman path integral
Operator theory
Quantum theory
Modified unitary form-sum Trotter–Kato product formula and Zeno product formula, which replace their short-time unitary groups appearing as factors in their products by their resolvents, are proved by Kato's method.
401
417
1
10.4171/175-1/19
http://www.ems-ph.org/doi/10.4171/175-1/19
Spectral asymptotics induced by approaching and diverging planar circles
Sylwia
Kondej
University of Zielona Góra, Poland
Schrödinger operator with delta potential, separation of variables, eigenvalues asymptotics
Partial differential equations
We consider two dimensional system governed by the Hamiltonian with delta interaction supported by two concentric circles separated by distance $d$. We analyze the asymptotics of the discrete eigenvalues for $d \to 0$ as well as for $d\to \infty$.
419
432
1
10.4171/175-1/20
http://www.ems-ph.org/doi/10.4171/175-1/20
Spectral estimates for the Heisenberg Laplacian on cylinders
Hynek
Kovařík
Università degli Studi di Brescia, Italy
Bartosch
Ruszkowski
Universität Stuttgart, Germany
Timo
Weidl
Universität Stuttgart, Germany
Heisenberg Laplacian, Berezin–Li–Yau inequality
Operator theory
Difference and functional equations
We study Riesz means of eigenvalues of the Heisenberg Laplacian with Dirichlet boundary conditions on a cylinder in dimension three. We obtain an inequality with a sharp leading term and an additional lower order term.
433
446
1
10.4171/175-1/21
http://www.ems-ph.org/doi/10.4171/175-1/21
Variational proof of the existence of eigenvalues for star graphs
Konstantin
Pankrashkin
Université Paris-Sud, Orsay, France
Singular Schrödinger operator, leaky graph, $\delta$-potential, star graph
Ordinary differential equations
We provide a purely variational proof of the existence of eigenvalues below the bottom of the essential spectrum for the Schrödinger operator with an attractive $\delta$-potential supported by a star graph, i.e. by a finite union of rays emanating from the same point. In contrast to the previous works, the construction is valid without any additional assumption on the number or the relative position of the rays. The approach is used to obtain an upper bound for the lowest eigenvalue.
447
458
1
10.4171/175-1/22
http://www.ems-ph.org/doi/10.4171/175-1/22
On the boundedness and compactness of weighted Green operators of second-order elliptic operators
Yehuda
Pinchover
Technion - Israel Institute of Technology, Haifa, Israel
Green function, ground state, Liouville theorem, positive solution, principal eigenvalue, small perturbation
Partial differential equations
Operator theory
For a given second-order linear elliptic operator $L$ which admits a positive minimal Green function, and a given positive weight function $W$, we introduce a family of weighted Lebesgue spaces $L^p(\phi_p)$ with their dual spaces, where $1\leq p\leq \infty$. We study some fundamental properties of the corresponding (weighted) Green operators on these spaces. In particular, we prove that these Green operators are bounded on $L^p(\phi_p)$ for any $1\leq p\leq \infty$ with a uniform bound. We study the existence of a principal eigenfunction for these operators in these spaces, and the simplicity of the corresponding principal eigenvalue. We also show that such a Green operator is a resolvent of a densely defined closed operator which is equal to $(-W^{-1})L$ on $C_0^\infty$, and that this closed operator generates a strongly continuous contraction semigroup. Finally, we prove that if $W$ is a (semi)small perturbation of $L$, then for any $1\leq p\leq \infty$, the associated Green operator is compact on $L^p(\phi_p)$, and the corresponding spectrum is $p$-independent.
459
489
1
10.4171/175-1/23
http://www.ems-ph.org/doi/10.4171/175-1/23
Abstract graph-like spaces and vector-valued metric graphs
Olaf
Post
Universität Trier, Germany
Abstract boundary value problems, Dirichlet-to-Neumann operator, graph Laplacians, coupled spaces
Operator theory
Combinatorics
Dynamical systems and ergodic theory
Global analysis, analysis on manifolds
In this note we present some abstract ideas how one can construct spaces from building blocks according to a graph. The coupling is expressed via boundary pairs, and can be applied to very different spaces such as discrete graphs, quantum graphs or graph-like manifolds. We show a spectral analysis of graph-like spaces, and consider as a special case vector-valued quantum graphs. Moreover, we provide a prototype of a convergence theorem for shrinking graph-like spaces with Dirichlet boundary conditions.
491
524
1
10.4171/175-1/24
http://www.ems-ph.org/doi/10.4171/175-1/24
A Cayley–Hamiltonian theorem for periodic finite band matrices
Barry
Simon
California Institute of Technology, Pasadena, USA
Periodic Jacobi matrices, discriminant, magic formula
Operator theory
Functions of a complex variable
Let $K$ be a doubly infinite, self-adjoint matrix which is finite band (i.e. $K_{jk} = 0$ if $|j-k| > m$) and periodic ($KS^n = S^nK$ for some $n$ where $(Su)_j = u_{j+1}$) and non-degenerate (i.e. $K_{j j+m} \ne 0$ for all $j$). Then, there is a polynomial, $p(x,y)$, in two variables with $p(K,S^n) = 0$. This generalizes the tridiagonal case where $p(x,y) = y^2 - y \Delta(x) + 1$ where $\Delta$ is the discriminant. I hope Pavel Exner will enjoy this birthday bouquet.
525
529
1
10.4171/175-1/25
http://www.ems-ph.org/doi/10.4171/175-1/25
Path topology dependence of adiabatic time evolution
Atushi
Tanaka
Tokyo Metropolitan University, Japan
Taksu
Cheon
Kochi University of Technology, Japan
Adiabatic theorem, eigenspace, homotopy, covering space
Quantum theory
Algebraic geometry
An adiabatic time evolution of a closed quantum system connects eigenspaces of initial and final Hermitian Hamiltonians for slowly driven systems, or, unitary Floquet operators for slowly modulated driven systems. We show that the connection of eigenspaces depends on a topological property of the adiabatic paths for given initial and final points. An example in slowly modulated periodically driven systems is shown. These analysis are based on the topological analysis of the exotic quantum holonomy in adiabatic closed paths.
531
542
1
10.4171/175-1/26
http://www.ems-ph.org/doi/10.4171/175-1/26
On quantum graph filters with flat passbands
Ondřej
Turek
Czech Academy of Sciences, Řež - Prague, Czech Republic
Quantum graph, vertex coupling, spectral filtering, quantum control
Quantum theory
We examine transmission through a quantum graph vertex to which auxiliary edges with constant potentials are attached. We find a characterization of vertex couplings for which the transmission probability from a given „input" line to a given „output" line shows a flat passband. The bandwidth is controlled directly by the potential on the auxiliary edges. Vertices with such couplings can thus serve as controllable band-pass filters. The paper extends earlier works on the topic. The result also demonstrates the effectivity of the $ST$-form of boundary conditions for a study of scattering in quantum graph vertices.
543
563
1
10.4171/175-1/27
http://www.ems-ph.org/doi/10.4171/175-1/27
Comments on the Chernoff $\sqrt n$-lemma
Valentin
Zagrebnov
Centre de Mathématiques et Informatique, Marseille, France
Chernoff lemma, semigroup theory, product formula, convergence rate
Operator theory
The Chernoff $\sqrt{n}$-lemma is revised. This concerns two aspects: an improvement of the Chernoff estimate in the strong operator topology and an operator-norm estimate for quasi-sectorial contractions. Applications to the Lie–Trotter product formula approximation for semigroups is presented.
565
573
1
10.4171/175-1/28
http://www.ems-ph.org/doi/10.4171/175-1/28
Frobenius Algebras II
Tilted and Hochschild Extension Algebras
Andrzej
Skowroński
Nicolaus Copernicus University, Toruń, Poland
Kunio
Yamagata
Tokyo University of Agriculture and Technology, Japan
Associative rings and algebras
Commutative rings and algebras
Linear and multilinear algebra; matrix theory
Category theory; homological algebra
(primary; secondary): 16-01; 13E10, 15A63, 15A69, 16Dxx, 16E10, 16E30, 16E40, 16G10, 16G20, 16G60, 16G70, 16S50, 16S70, 18A25, 18E30, 18G15
Mathematics
Algebra
Algebra, module, bimodule, representation, quiver, ideal, radical, simple module, semisimple module, uniserial module, projective module, injective module, tilting module, hereditary algebra, tilted algebra, Frobenius algebra, symmetric algebra, selfinjective algebra, Hochschild extension algebra, category, functor, torsion pair, projective dimension, injective dimension, global dimension, Euler form, Grothendieck group, irreducible homomorphism, almost split sequence, Auslander-Reiten translation, Auslander-Reiten quiver, stable equivalence, syzygy module, duality bimodule, Hochschild extension
This is the second of three volumes which will provide a comprehensive introduction to the modern representation theory of Frobenius algebras. The first part of the book is devoted to fundamental results of the representation theory of finite dimensional hereditary algebras and their tilted algebras, which allow to describe the representation theory of prominent classes of Frobenius algebras. The second part is devoted to basic classical and recent results concerning the Hochschild extensions of finite dimensional algebras by duality bimodules and their module categories. Moreover, the shapes of connected components of the stable Auslander-Reiten quivers of Frobenius algebras are described. The only prerequisite in this volume is a basic knowledge of linear algebra and some results of the fi rst volume. It includes complete proofs of all results presented and provides a rich supply of examples and exercises. The text is primarily addressed to graduate students starting research in the representation theory of algebras as well mathematicians working in other fi elds.
5
12
2017
978-3-03719-174-3
978-3-03719-674-8
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/174
http://www.ems-ph.org/doi/10.4171/174
EMS Textbooks in Mathematics
Asymptotic Theory of Transaction Costs
Walter
Schachermayer
Universität Wien, Austria
Statistics
Probability theory and stochastic processes
Game theory, economics, social and behavioral sciences
Primary: 62P05, 91G10; Secondary: 60G44
Probability + statistics
Portfolio optimization, transaction costs, shadow price, semimartingale, fractional Brownian motion
A classical topic in Mathematical Finance is the theory of portfolio optimization. Robert Merton's work from the early seventies had enormous impact on academic research as well as on the paradigms guiding practitioners. One of the ramifications of this topic is the analysis of (small) proportional transaction costs, such as a Tobin tax. The lecture notes present some striking recent results of the asymptotic dependence of the relevant quantities when transaction costs tend to zero. An appealing feature of the consideration of transaction costs is that it allows for the first time to reconcile the no arbitrage paradigm with the use of non-semimartingale models, such as fractional Brownian motion. This leads to the culminating theorem of the present lectures which roughly reads as follows: for a fractional Brownian motion stock price model we always find a shadow price process for given transaction costs. This process is a semimartingale and can therefore be dealt with using the usual machinery of mathematical finance.
3
10
2017
978-3-03719-173-6
978-3-03719-673-1
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/173
http://www.ems-ph.org/doi/10.4171/173
Zurich Lectures in Advanced Mathematics
PDE Models for Chemotaxis and Hydrodynamics in Supercritical Function Spaces
Hans
Triebel
University of Jena, Germany
Partial differential equations
35–02, 46–02, 76–02, 92–02; 35K05, 35Q30, 35Q92, 42B35, 46E35, 76D05, 92C15, 92C17
Differential equations
Function spaces of Besov–Sobolev type, chemotaxis, hydrodynamics, heat equations, Keller–Segel equations, Navier–Stokes equations
This book deals with PDE models for chemotaxis (the movement of biological cells or organisms in response of chemical gradients) and hydrodynamics (viscous, homogeneous, and incompressible fluid filling the entire space). The underlying Keller–Segel equations (chemotaxis), Navier–Stokes equations (hydrodynamics), and their numerous modifications and combinations are treated in the context of inhomogeneous spaces of Besov–Sobolev type paying special attention to mapping properties of related nonlinearities. Further models are considered, including (deterministic) Fokker–Planck equations and chemotaxis Navier–Stokes equations. These notes are addressed to graduate students and mathematicians having a working knowledge of basic elements of the theory of function spaces, especially of Besov-Sobolev type and interested in mathematical biology and physics.
3
23
2017
978-3-03719-172-9
978-3-03719-672-4
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/172
http://www.ems-ph.org/doi/10.4171/172
EMS Series of Lectures in Mathematics
2523-5176
2523-5184
Representation Theory – Current Trends and Perspectives
Henning
Krause
Universität Bielefeld, Germany
Peter
Littelmann
Universität Köln, Germany
Gunter
Malle
Universität Kaiserslautern, Germany
Karl-Hermann
Neeb
FAU Erlangen-Nürnberg, Germany
Christoph
Schweigert
Universität Hamburg, Germany
Algebraic geometry
Associative rings and algebras
Nonassociative rings and algebras
Category theory; homological algebra
Primary: 14Mxx,16Gxx, 17Bxx, 18Exx, 20Gxx, 22Exx; secondary: 58Cxx, 81Txx.
Algebraic geometry
Algebraic groups, bounded and semibounded representations, categorification, character formulae, cluster algebras, Deligne-Lusztig theory, flat degenerations, geometrization, higher representation theory, highest weight categories, infinite dimensional Lie groups, local-global conjectures, special varieties, topological field theory
From April 2009 until March 2016, the German Science Foundation supported generously the Priority Program SPP 1388 in Representation Theory. The core principles of the projects realized in the framework of the priority program have been categorification and geometrization, this is also reflected by the contributions to this volume. Apart from the articles by former postdocs supported by the priority program, the volume contains a number of invited research and survey articles, many of them are extended versions of talks given at the last joint meeting of the priority program in Bad Honnef in March 2015. This volume is covering current research topics from the representation theory of finite groups, of algebraic groups, of Lie superalgebras, of finite dimensional algebras and of infinite dimensional Lie groups. Graduate students and researchers in mathematics interested in representation theory will find this volume inspiring. It contains many stimulating contributions to the development of this broad and extremely diverse subject.
1
12
2017
978-3-03719-171-2
978-3-03719-671-7
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/171
http://www.ems-ph.org/doi/10.4171/171
EMS Series of Congress Reports
2523-515X
2523-5168
Symmetric superspaces: slices, radial parts, and invariant functions
Alexander
Alldridge
Universität Köln, Germany
Chevalley restriction theorem, differential operator, Harish-Chandra homomorphism, Lie superalgebra, radial part, Riemannian symmetric superspace
Global analysis, analysis on manifolds
Nonassociative rings and algebras
Differential geometry
General
We study the restriction of invariant polynomials on the tangent space of a Riemannian symmetric supermanifold to a Cartan subspace. We survey known results in the case the symmetric space is a Lie supergroup, and more generally, where the Cartan subspace is even. We then describe an approach to this problem, developed in joint work in progress with K. Coulembier, based on the study of radial parts of di fferential operators. This leads to a characterisation of the invariant functions for an arbitrary linear isometric action, and as a special case, to a Chevalley restriction theorem valid for the isotropy representation of any contragredient Riemannian symmetric superspace.
1
11
1
10.4171/171-1/1
http://www.ems-ph.org/doi/10.4171/171-1/1
Geometry of quiver Grassmannians of Dynkin type with applications to cluster algebras
Giovanni
Cerulli Irelli
Università di Roma La Sapienza, Italy
Quiver Grassmannians, Dynkin quivers, cluster algebras
Associative rings and algebras
Algebraic geometry
General
The paper includes a new proof of the fact that quiver Grassmannians associated with rigid representations of Dynkin quivers do not have cohomology in odd degrees. Moreover, it is shown that they do not have torsion in homology. A new proof of the Caldero–Chapoton formula is provided. As a consequence a new proof of the positivity of cluster monomials in the acyclic clusters associated with Dynkin quivers is obtained. The methods used here are based on joint works with Markus Reineke and Evgeny Feigin.
13
45
1
10.4171/171-1/2
http://www.ems-ph.org/doi/10.4171/171-1/2
Spherical varieties and perspectives in representation theory
Stéphanie
Cupit-Foutou
Ruhr-Universität Bochum, Germany
Reductive groups, spherical varieties
Algebraic geometry
Group theory and generalizations
General
This paper is a brief overview on recent classi cation results and related problems concerning spherical varieties. The emphasis is made on the representation theoretical aspects of these objects.
47
57
1
10.4171/171-1/3
http://www.ems-ph.org/doi/10.4171/171-1/3
Categorical actions from Lusztig induction and restriction on finite general linear groups
Olivier
Dudas
Université Paris Diderot Paris 7, France
Michela
Varagnolo
Université de Cergy-Pontoise, France
Éric
Vasserot
Université Paris Diderot Paris 7, France
Finite reductive groups, Deligne–Lusztig theory, higher representation theory
Group theory and generalizations
General
In this note we explain how Lusztig's induction and restriction functors yield categorical actions of Kac–Moody algebras on the derived category of unipotent representations. We focus on the example of finite general linear groups and induction/restriction associated with split Levi subgroups, providing a derived analogue of Harish–Chandra induction/restriction as studied by Chuang–Rouquier in [5].
59
74
1
10.4171/171-1/4
http://www.ems-ph.org/doi/10.4171/171-1/4
Homological mirror symmetry for singularities
Wolfgang
Ebeling
Universität Hannover, Germany
Homological mirror symmetry, singularities, strange duality, invertible polynomials, derived categories, weighted projective lines, Coxeter-Dynkin diagrams, group action, orbifold E-function, Burnside ring, unimodal, bimodal
Algebraic geometry
Associative rings and algebras
Several complex variables and analytic spaces
Differential geometry
We give a survey on results related to the Berglund–Hubsch duality of invertible polynomials and the homological mirror symmetry conjecture for singularities.
75
107
1
10.4171/171-1/5
http://www.ems-ph.org/doi/10.4171/171-1/5
On the category of finite-dimensional representations of OSp$(r|2n)$: Part I
Michael
Ehrig
Universität Bonn, Germany
Catharina
Stroppel
Universität Bonn, Germany
Orthosymplectic supergroup, finite dimensional representations, Brauer algebra, Deligne category
Nonassociative rings and algebras
General
We study the combinatorics of the category $\mathcal F$ of fi nite-dimensional modules for the orthosymplectic Lie supergroup OSp$(r|2n)$. In particular we present a positive counting formula for the dimension of the space of homomorphisms between two projective modules. This refi nes earlier results of Gruson and Serganova. For each block $\mathcal B$ we construct an algebra $A_\mathcal B$ whose module category shares the combinatorics with $\mathcal B$. It arises as a subquotient of a suitable limit of type D Khovanov algebras. It turns out that $A_\mathcal B$ is isomorphic to the endomorphism algebra of a minimal projective generator of $\mathcal B$. This provides a direct link from $\mathcal F$ to parabolic categories $\mathcal O$ of type B/D, with maximal parabolic of type A, to the geometry of isotropic Grassmannians of types B/D and to Springer fi bres of type C/D. We also indicate why $\mathcal F$ is not highest weight in general.
109
170
1
10.4171/171-1/6
http://www.ems-ph.org/doi/10.4171/171-1/6
On cubes of Frobenius extensions
Ben
Elias
University of Oregon, Eugene, USA
Noah
Snyder
Indiana University, Bloomington, United States
Geordie
Williamson
Max-Planck-Institut für Mathematik, Bonn, Germany
Frobenius extensions, diagrammatic algebra, Soergel bimodules
$K$-theory
Nonassociative rings and algebras
Category theory; homological algebra
General
Given a hypercube of Frobenius extensions between commutative algebras, we provide a diagrammatic description of some natural transformations between compositions of induction and restriction functors, in terms of colored transversely-intersecting planar 1-manifolds. The relations arise in the first and third author’s work on (singular) Soergel bimodules.
171
186
1
10.4171/171-1/7
http://www.ems-ph.org/doi/10.4171/171-1/7
On toric degenerations of flag varieties
Xin
Fang
Universität Köln, Germany
Ghislain
Fourier
University of Glasgow, UK
Peter
Littelmann
Universität Köln, Germany
Flag varieties, spherical varieties, cluster algebras, toric degenerations
Algebraic geometry
Convex and discrete geometry
General
Following the historical track in pursuing $T$-equivariant flat toric degenerations of flag varieties and spherical varieties, we explain how powerful tools in algebraic geometry and representation theory, such as canonical bases, Newton–Okounkov bodies, PBW- filtrations and cluster algebras come to push the subject forward.
187
232
1
10.4171/171-1/8
http://www.ems-ph.org/doi/10.4171/171-1/8
Subquotient categories of the affine category $\mathcal O$ at the critical level
Peter
Fiebig
Universität Erlangen-Nürnberg, Germany
Restricted representations, critical level, Kac–Moody algebras, Feigin–Frenkel conjecture
Nonassociative rings and algebras
Quantum theory
General
We introduce subquotient categories of the restricted category $\mathcal O$ over an affi ne Kac–Moody algebra at the critical level and show that some of them have a realization in terms of moment graph sheaves.
233
253
1
10.4171/171-1/9
http://www.ems-ph.org/doi/10.4171/171-1/9
Low-dimensional topology, low-dimensional field theory and representation theory
Jürgen
Fuchs
Karlstads Universitet, Sweden
Christoph
Schweigert
Universität Hamburg, Germany
Topological field theory, tensor categories, categorification
Quantum theory
Manifolds and cell complexes
General
Structures in low-dimensional topology and low-dimensional geometry – often combined with ideas from (quantum) fi eld theory – can explain and inspire concepts in algebra and in representation theory and their categori ed versions. We present a personal view on some of these instances which have appeared within the Research Priority Programme SPP 1388 "Representation theory".
255
267
1
10.4171/171-1/10
http://www.ems-ph.org/doi/10.4171/171-1/10
Derived categories of quasi-hereditary algebras and their derived composition series
Martin
Kalck
University of Edinburgh, UK
Triangulated categories, quasi-hereditary algebras, exceptional sequences, recollements, gentle algebras, derived equivalences, derived composition series, derived Jordan–Hölder property
Category theory; homological algebra
Associative rings and algebras
General
We study composition series of derived module categories in the sense of Angeleri Hügel, König & Liu for quasi-hereditary algebras. More precisely, we show that having a composition series with all factors being derived categories of vector spaces does not characterise derived categories of quasi-hereditay algebras. This gives a negative answer to a question of Liu & Yang and the proof also confi rms part of a conjecture of Bobi nski & Malicki. In another direction, we show that derived categories of quasi-hereditary algebras can have composition series with lots of diff erent lengths and composition factors. In other words, there is no Jordan–Hölder property for composition series of derived categories of quasi-hereditary algebras.
269
308
1
10.4171/171-1/11
http://www.ems-ph.org/doi/10.4171/171-1/11
Dominant dimension and applications
Steffen
Koenig
Universität Stuttgart, Germany
Dominant dimension, representation dimension, Schur algebras, gendosymmetric algebras
Associative rings and algebras
General
Dominant dimension is a little known homological dimension, which is, however, crucial in many respects, both for abstractly studying fi nite dimensional algebras and their representation theory, and for applications to group algebras or in algebraic Lie theory. Various aspects and recent applications of dominant dimension will be outlined and illustrated.
309
330
1
10.4171/171-1/12
http://www.ems-ph.org/doi/10.4171/171-1/12
Highest weight categories and strict polynomial functors. With an appendix by Cosima Aquilino
Henning
Krause
Universität Bielefeld, Germany
Highest weight category, strict polynomial functor, polynomial representation, divided power, Schur algebra, quasi-hereditary algebra, Ringel duality
Group theory and generalizations
Category theory; homological algebra
General
Highest weight categories are described in terms of standard objects and recollements of abelian categories, working over an arbitrary commutative base ring. Then the highest weight structure for categories of strict polynomial functors is explained, using the theory of Schur and Weyl functors. A consequence is the well-known fact that Schur algebras are quasi-hereditary.
331
373
1
10.4171/171-1/13
http://www.ems-ph.org/doi/10.4171/171-1/13
In the bocs seat: Quasi-hereditary algebras and representation type
Julian
Külshammer
Universität Stuttgart, Germany
BGG category $\mathcal O$, bocs, Eilenberg–Moore category, exact Borel subalgebra, Kleisli category, $q$-Schur algebras, quasi-hereditary algebras, reduction algorithm, representation type, Schur algebras, tame, wild
Associative rings and algebras
Nonassociative rings and algebras
Category theory; homological algebra
General
This paper surveys bocses, quasi-hereditary algebras and their relationship which was established in a recent result by Koenig, Ovsienko, and the author. Particular emphasis is placed on applications of this result to the representation type of the category filtered by standard modules for a quasi-hereditary algebra. In this direction, joint work with Thiel is presented showing that the subcategory of modules fi ltered by Weyl modules for tame Schur algebras is of fi nite representation type. The paper also includes a new proof for the classi cation of quasi-hereditary algebras with two simple modules, a result originally obtained by Membrillo–Hernández in [70].
375
426
1
10.4171/171-1/14
http://www.ems-ph.org/doi/10.4171/171-1/14
From groups to clusters
Sefi
Ladkani
University of Haifa, Israel
Algebra of quaternion type, 2-CY-tilted algebra, Brauer graph algebra, derived equivalence, Jacobian algebra, marked surface, periodic modules, quiver with potential, ribbon graph, ribbon quiver, triangulation quiver, triangulation algebra, symmetric algebra
Associative rings and algebras
Commutative rings and algebras
Category theory; homological algebra
Group theory and generalizations
We construct a new class of symmetric algebras of tame representation type that are also the endomorphism algebras of cluster-tilting objects in 2-Calabi–Yau triangulated categories, hence all their non-projective indecomposable modules are $\Omega$-periodic of period dividing 4. Our construction is based on the combinatorial notion of triangulation quivers, which arise naturally from triangulations of oriented surfaces with marked points. This class of algebras contains the algebras of quaternion type introduced and studied by Erdmann with relation to certain blocks of group algebras. On the other hand, it contains also the Jacobian algebras of the quivers with potentials associated by Fomin–Shapiro–Thurston and Labardini–Fragoso to triangulations of closed surfaces with punctures, hence our construction may serve as a bridge between the modular representation theory of nite groups and the theory of cluster algebras.
427
500
1
10.4171/171-1/15
http://www.ems-ph.org/doi/10.4171/171-1/15
Semi-infinite combinatorics in representation theory
Martina
Lanini
Edinburgh University, UK
Moment graphs, semi-infinite order, character formulae
Nonassociative rings and algebras
Group theory and generalizations
General
In this work we discuss some appearances of semi-infi nite combinatorics in representation theory. We propose a semi-in finite moment graph theory and we motivate it by considering the (not yet rigorously de fined) geometric side of the story. We show that it is possible to compute stalks of the local intersection cohomology of the semi-infi nite flag variety, and hence of spaces of quasi maps, by performing an algorithm due to Braden and MacPherson.
501
518
1
10.4171/171-1/16
http://www.ems-ph.org/doi/10.4171/171-1/16
Local-global conjectures in the representation theory of finite groups
Gunter
Malle
TU Kaiserslautern, Germany
Local-global conjectures, McKay conjecture, Alperin–McKay conjecture, Alperin weight conjecture, Brauer's height zero conjecture, Dade conjecture, reduction theorems
Group theory and generalizations
General
We give a survey of recent developments in the investigation of the various local-global conjectures for representations of nite groups.
519
539
1
10.4171/171-1/17
http://www.ems-ph.org/doi/10.4171/171-1/17
Bounded and semibounded representations of infinite dimensional Lie groups
Karl-Hermann
Neeb
Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
Infinite dimensional Lie group, host algebra, semibounded representation, bounded representation, holomorphic induction
Topological groups, Lie groups
General
In this note we describe the recent progress in the classi fication of bounded and semibounded representations of in finite dimensional Lie groups. We start with a discussion of the semiboundedness condition and how the new concept of a smoothing operator can be used to construct $C*$ -algebras (so called host algebras) whose representations are in one-to-one correspondence with certain semibounded representations of an infi nite dimensional Lie group $G$. This makes the full power of $C*$ -theory available in this context. Then we discuss the classi cation of bounded representations of several types of unitary groups on Hilbert spaces and of gauge groups. After explaining the method of holomorphic induction as a means to pass from bounded representations to semibounded ones, we describe the classifi cation of semibounded representations for hermitian Lie groups of operators, loop groups (with infi nite dimensional targets), the Virasoro group and certain in finite dimensional oscillator groups.
541
563
1
10.4171/171-1/18
http://www.ems-ph.org/doi/10.4171/171-1/18
On ideals in $\operatorname{U}(\mathfrak {sl} (\infty)), \operatorname{U}(\mathfrak {o} (\infty)), \operatorname{U}(\mathfrak {sp} (\infty))$
Ivan
Penkov
Jacobs-Universität Bremen, Germany
Alexey
Petukhov
The University of Manchester, UK
Primitive ideals, finitary Lie algebras, highest weight modules, $\mathfrak{osp}$-duality
Nonassociative rings and algebras
General
We provide a review of results on two-sided ideals in the enveloping algebra $\operatorname{U}(\mathfrak g(\infty))$ of a locally simple Lie algebra $\mathfrak g(\infty)$. We pay special attention to the case when $\mathfrak g(\infty)$ is one of the finitary Lie algebras $\mathfrak{sl}(\infty), \mathfrak o(\infty), \mathfrak{sp}(\infty)$. The main results include a description of all integrable ideals in $\operatorname{U}(\mathfrak g(\infty))$, as well as a criterion for the annihilator of an arbitrary (not necessarily integrable) simple highest weight module to be nonzero. This criterion is new for $\mathfrak g(\infty)=\mathfrak o(\infty), \mathfrak{sp}(\infty)$. All annihilators of simple highest weight modules are integrable ideals for $\mathfrak g(\infty)=\mathfrak{sl}(\infty),$ $\mathfrak o(\infty)$. Finally, we prove that the lattices of ideals in $\operatorname{U}(\mathfrak o(\infty))$ and $\operatorname{U}(\mathfrak{sp}(\infty))$ are isomorphic.
565
602
1
10.4171/171-1/19
http://www.ems-ph.org/doi/10.4171/171-1/19
Spherical varieties: applications and generalizations
Guido
Pezzini
Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
Algebraic groups, spherical varieties, representation theory, Kac–Moody groups
Algebraic geometry
Group theory and generalizations
General
In this short note we review some applications of the theory of spherical varieties in related fields, some generalizations of this theory, and present some open problems.
603
612
1
10.4171/171-1/20
http://www.ems-ph.org/doi/10.4171/171-1/20
Quiver moduli and small desingularizations of some GIT quotients
Markus
Reineke
Ruhr-Universität Bochum, Germany
Quiver moduli, GIT quotients, small desingularizations
Associative rings and algebras
Algebraic geometry
Algebraic topology
General
We construct small desingularizations of moduli spaces of semistable quiver representations for indivisible dimension vectors using deformations of stabilities and a dimension estimate for nullcones. We apply this construction to several classes of GIT quotients.
613
635
1
10.4171/171-1/21
http://www.ems-ph.org/doi/10.4171/171-1/21
Geometric invariant theory for principal three-dimensional subgroups acting on flag varieties
Henrik
Seppänen
Georg-August Universität Göttingen, Germany
Valdemar
Tsanov
Georg-August Universität Göttingen, Germany
Flag variety, geometric invariant theory, principal SL(2)-subgroup, branching cone, Mori dream space
Algebraic geometry
Nonassociative rings and algebras
General
Let $G$ be a semisimple complex Lie group. In this article, we study Geometric Invariant Theory on a flag variety $G/B$ with respect to the action of a principal 3-dimensional simple subgroup $S\subset G$. We determine explicitly the GIT-equivalence classes of $S$-ample line bundles on $G/B$. We show that, under mild assumptions, among the GIT-classes there are chambers, in the sense of Dolgachev-Hu. The GIT-quotients with respect to various chambers form a family of Mori dream spaces, canonically associated with $G$. We are able to determine the three important cones in the Picard group of any of these quotients: the pseudoeffective-, the movable-, and the nef cones.
637
663
1
10.4171/171-1/22
http://www.ems-ph.org/doi/10.4171/171-1/22
Inductive conditions for counting conjectures via character triples
Britta
Späth
Bergische Universität Wuppertal, Germany
Reduction theorems, character triples, counting conjectures
Group theory and generalizations
General
In recent years several global/local conjectures in the representation theory of fi nite groups have been reduced to conditions on quasi-simple groups. We reformulate the inductive conditions for the conjectures by Alperin and McKay using (new) order relations between ordinary, respectively modular character triples. This allows to clarify the similarities and di fferences between those conditions.
665
680
1
10.4171/171-1/23
http://www.ems-ph.org/doi/10.4171/171-1/23
Restricted rational Cherednik algebras
Ulrich
Thiel
Universität Stuttgart, Germany
Rational Cherednik algebras, Calogero–Moser spaces, reflection groups
Associative rings and algebras
Group theory and generalizations
General
We give an overview of the representation theory of restricted rational Cherednik algebras. These are certain finite-dimensional quotients of rational Cherednik algebras at $t = 0$. Their representation theory is connected to the geometry of the Calogero–Moser space, and there is a lot of evidence that they contain certain information about Hecke algebras even though the precise connection is so far unclear. We outline the basic theory along with some open problems and conjectures, and give explicit results in the cyclic and dihedral cases.
681
745
1
10.4171/171-1/24
http://www.ems-ph.org/doi/10.4171/171-1/24
On the existence of regular vectors
Christoph
Zellner
Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
infinite dimensional Lie group, smooth vector, analytic vector, semibounded representation, continuous representation
Topological groups, Lie groups
General
Let $G$ be a locally convex Lie group and $\pi:G \to \mathrm{U}(\mathcal H)$ be a continuous unitary representation. $\pi$ is called smooth if the space of $\pi$-smooth vectors $\mathcal H^\infty\subset \mathcal H$ is dense. In this article we show that under certain conditions, concerning in particular the structure of the Lie algebra $\mathfrak{g}$ of $G$, a continuous unitary representation of $G$ is automatically smooth. As an application, this yields a dense space of smooth vectors for continuous positive energy representations of oscillator groups, double extensions of loop groups and the Virasoro group. Moreover we show the existence of a dense space of analytic vectors for the class of semibounded representations of Banach–Lie groups. Here $\pi$ is called semibounded, if $\pi$ is smooth and there exists a non-empty open subset $U\subset\mathfrak{g}$ such that the operators $i \mathrm d\ pi(x)$ from the derived representation are uniformly bounded from above for $x \in U$.
747
763
1
10.4171/171-1/25
http://www.ems-ph.org/doi/10.4171/171-1/25
The Monge–Ampère Equation and Its Applications
Alessio
Figalli
ETH Zürich, Switzerland
Partial differential equations
Differential geometry
Primary: 35J96; Secondary: 35B65, 35J60, 35J66, 35B45, 35B50, 35D05, 35D10, 35J65, 53A15, 53C45
Differential equations
Differential + Riemannian geometry
Monge–Ampère equation, weak and strong solutions, existence, uniqueness, regularity
The Monge–Ampère equation is one of the most important partial differential equations, appearing in many problems in analysis and geometry. This monograph is a comprehensive introduction to the existence and regularity theory of the Monge–Ampère equation and some selected applications; the main goal is to provide the reader with a wealth of results and techniques he or she can draw from to understand current research related to this beautiful equation. The presentation is essentially self-contained, with an appendix wherein one can find precise statements of all the results used from different areas (linear algebra, convex geometry, measure theory, nonlinear analysis, and PDEs). This book is intended for graduate students and researchers interested in nonlinear PDEs: explanatory figures, detailed proofs, and heuristic arguments make this book suitable for self-study and also as a reference.
1
1
2017
978-3-03719-170-5
978-3-03719-670-0
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/170
http://www.ems-ph.org/doi/10.4171/170
Zurich Lectures in Advanced Mathematics
Bound States of the Magnetic Schrödinger Operator
Nicolas
Raymond
Université de Rennes, France
Partial differential equations
Calculus of variations and optimal control; optimization
Quantum theory
35P15, 35P20, 49R05, 81Q10, 81Q20
Differential equations
Mathematical logic
Magnetic Schrödinger equation, discrete spectrum, semiclassical analysis, magnetic harmonic approximation
This book is a synthesis of recent advances in the spectral theory of the magnetic Schrödinger operator. It can be considered a catalog of concrete examples of magnetic spectral asymptotics. Since the presentation involves many notions of spectral theory and semiclassical analysis, it begins with a concise account of concepts and methods used in the book and is illustrated by many elementary examples. Assuming various points of view (power series expansions, Feshbach–Grushin reductions, WKB constructions, coherent states decompositions, normal forms) a theory of Magnetic Harmonic Approximation is then established which allows, in particular, accurate descriptions of the magnetic eigenvalues and eigenfunctions. Some parts of this theory, such as those related to spectral reductions or waveguides, are still accessible to advanced students while others (e.g., the discussion of the Birkhoff normal form and its spectral consequences, or the results related to boundary magnetic wells in dimension three) are intended for seasoned researchers.
1
1
2017
978-3-03719-169-9
978-3-03719-669-4
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/169
http://www.ems-ph.org/doi/10.4171/169
EMS Tracts in Mathematics
27
Dynamics Done with Your Bare Hands
Lecture notes by Diana Davis, Bryce Weaver, Roland K. W. Roeder, Pablo Lessa
Françoise
Dal’Bo
Université de Rennes I, France
François
Ledrappier
University of Notre Dame, USA
Amie
Wilkinson
University of Chicago, USA
Dynamical systems and ergodic theory
Differential geometry
37A, 37B, 37D,37F, 37H, 53A
Differential equations
Dynamical systems, geometry, ergodic theory, billards, complex dynamics, random walk, group theory
This book arose from 4 lectures given at the Undergraduate Summer School of the Thematic Program Dynamics and Boundaries held at the University of Notre Dame. It is intended to introduce (under)graduate students to the field of dynamical systems by emphasizing elementary examples, exercises and bare hands constructions. The lecture of Diana Davis is devoted to billiard flows on polygons, a simple-sounding class of continuous time dynamical system for which many problems remain open. Bryce Weaver focuses on the dynamics of a 2x2 matrix acting on the flat torus. This example introduced by Vladimir Arnold illustrates the wide class of uniformly hyperbolic dynamical systems, including the geodesic flow for negatively curved, compact manifolds. Roland Roeder considers a dynamical system on the complex plane governed by a quadratic map with a complex parameter. These maps exhibit complicated dynamics related to the Mandelbrot set defined as the set of parameters for which the orbit remains bounded. Pablo Lessa deals with a type of non-deterministic dynamical system: a simple walk on an infinite graph, obtained by starting at a vertex and choosing a random neighbor at each step. The central question concerns the recurrence property. When the graph is a Cayley graph of a group, the behavior of the walk is deeply related to algebraic properties of the group.
11
30
2016
978-3-03719-168-2
978-3-03719-668-7
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/168
http://www.ems-ph.org/doi/10.4171/168
EMS Series of Lectures in Mathematics
2523-5176
2523-5184
Lines in positive genus: An introduction to flat surfaces
Diana
Davis
Williams College, Williamstown, USA
Dynamical systems and ergodic theory
General
1
55
1
10.4171/168-1/1
http://www.ems-ph.org/doi/10.4171/168-1/1
Introduction to complicated behavior and periodic orbits
Bryce
Weaver
James Madison University, Harrisonburg, USA
Dynamical systems and ergodic theory
General
57
100
1
10.4171/168-1/2
http://www.ems-ph.org/doi/10.4171/168-1/2
Around the boundary of complex dynamics
Roland
Roeder
Indiana University Purdue University Indianapolis, USA
Dynamical systems and ergodic theory
General
101
155
1
10.4171/168-1/3
http://www.ems-ph.org/doi/10.4171/168-1/3
Recurrence vs transience: An introduction to random walks
Pablo
Lessa
Universidad de la República, Montevideo, Uruguay
Dynamical systems and ergodic theory
General
157
201
1
10.4171/168-1/4
http://www.ems-ph.org/doi/10.4171/168-1/4
Degenerate Complex Monge–Ampère Equations
Vincent
Guedj
Université Paul Sabatier, Toulouse, France
Ahmed
Zeriahi
Université Paul Sabatier, Toulouse, France
Several complex variables and analytic spaces
32W20, 32U20, 32Q20, 35D30, 53C55, 32U15, 32U40, 35B65
Complex analysis
Pluripotential theory, plurisubharmonic functions, complex Monge–Ampère operators, generalized capacities, weak solutions, a priori estimates, canonical Kähler metrics, singular varieties
Winner of the 2016 EMS Monograph Award! Complex Monge–Ampère equations have been one of the most powerful tools in Kähler geometry since Aubin and Yau’s classical works, culminating in Yau’s solution to the Calabi conjecture. A notable application is the construction of Kähler-Einstein metrics on some compact Kähler manifolds. In recent years degenerate complex Monge–Ampère equations have been intensively studied, requiring more advanced tools. The main goal of this book is to give a self-contained presentation of the recent developments of pluripotential theory on compact Kähler manifolds and its application to Kähler–Einstein metrics on mildly singular varieties. After reviewing basic properties of plurisubharmonic functions, Bedford–Taylor’s local theory of complex Monge–Ampère measures is developed. In order to solve degenerate complex Monge–Ampère equations on compact Kähler manifolds, fine properties of quasi-plurisubharmonic functions are explored, classes of finite energies defined and various maximum principles established. After proving Yau’s celebrated theorem as well as its recent generalizations, the results are then used to solve the (singular) Calabi conjecture and to construct (singular) Kähler–Einstein metrics on some varieties with mild singularities. The book is accessible to advanced students and researchers of complex analysis and differential geometry.
1
12
2017
978-3-03719-167-5
978-3-03719-667-0
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/167
http://www.ems-ph.org/doi/10.4171/167
EMS Tracts in Mathematics
26
Metric Geometry of Locally Compact Groups
Yves
Cornulier
Université Paris-Sud, Orsay, France
Pierre
de la Harpe
Université de Genève, Switzerland
Group theory and generalizations
Topological groups, Lie groups
Geometry
Manifolds and cell complexes
Primary: 20F65; Secondary: 20F05, 22D05, 51F99, 54E35, 57M07, 57T20
Groups + group theory
Locally compact groups, left-invariant metrics, $\sigma$-compactness, second countability, compact generation, compact presentation, metric coarse equivalence, quasi-isometry, coarse connectedness, coarse simple connectedness, growth, amenability
Winner of the 2016 EMS Monograph Award! The main aim of this book is the study of locally compact groups from a geometric perspective, with an emphasis on appropriate metrics that can be defined on them. The approach has been successful for finitely generated groups, and can favourably be extended to locally compact groups. Parts of the book address the coarse geometry of metric spaces, where ‘coarse’ refers to that part of geometry concerning properties that can be formulated in terms of large distances only. This point of view is instrumental in studying locally compact groups. Basic results in the subject are exposed with complete proofs, others are stated with appropriate references. Most importantly, the development of the theory is illustrated by numerous examples, including matrix groups with entries in the the field of real or complex numbers, or other locally compact fields such as p-adic fields, isometry groups of various metric spaces, and, last but not least, discrete group themselves. The book is aimed at graduate students and advanced undergraduate students, as well as mathematicians who wish some introduction to coarse geometry and locally compact groups.
9
30
2016
978-3-03719-166-8
978-3-03719-666-3
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/166
http://www.ems-ph.org/doi/10.4171/166
EMS Tracts in Mathematics
25
Free Probability and Operator Algebras
Dan-Virgil
Voiculescu
University of California, Berkeley, USA
Nicolai
Stammeier
University of Oslo, Norway
Moritz
Weber
Universität Saarbrücken, Germany
Functional analysis
Group theory and generalizations
Operator theory
Probability theory and stochastic processes
Primary 46L54; secondary 60B20, 47C15, 20G42
Functional analysis
Groups + group theory
Free probability, operator algebras, random matrices, free monotone transport, free group factors, free convolution, compact quantum groups, easy quantum groups, noncrossing partitions, free independence, entropy, max-stable laws, exchangeability
Free probability is a probability theory dealing with variables having the highest degree of noncommutativity, an aspect found in many areas (quantum mechanics, free group algebras, random matrices etc). Thirty years after its foundation, it is a well-established and very active field of mathematics. Originating from Voiculescu’s attempt to solve the free group factor problem in operator algebras, free probability has important connections with random matrix theory, combinatorics, harmonic analysis, representation theory of large groups, and wireless communication. These lecture notes arose from a masterclass in Münster, Germany and present the state of free probability from an operator algebraic perspective. This volume includes introductory lectures on random matrices and combinatorics of free probability (Speicher), free monotone transport (Shlyakhtenko), free group factors (Dykema), free convolution (Bercovici), easy quantum groups (Weber), and a historical review with an outlook (Voiculescu). In order to make it more accessible, the exposition features a chapter on basics in free probability, and exercises for each part. This book is aimed at master students to early career researchers familiar with basic notions and concepts from operator algebras.
7
31
2016
978-3-03719-165-1
978-3-03719-665-6
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/165
http://www.ems-ph.org/doi/10.4171/165
Münster Lectures in Mathematics
2523-5230
2523-5249
Background and outlook
Dan-Virgil
Voiculescu
University of California, Berkeley, United States
Functional analysis
General
1
6
1
10.4171/165-1/1
http://www.ems-ph.org/doi/10.4171/165-1/1
Basics in free probability
Moritz
Weber
Universität des Saarlandes, Saarbrücken, Germany
Probability theory and stochastic processes
General
7
16
1
10.4171/165-1/2
http://www.ems-ph.org/doi/10.4171/165-1/2
Random matrices and combinatorics
Roland
Speicher
Universität des Saarlandes, Saarbrücken, Germany
Probability theory and stochastic processes
General
17
37
1
10.4171/165-1/3
http://www.ems-ph.org/doi/10.4171/165-1/3
Free monotone transport
Dimitri
Shlyakhtenko
University of California Los Angeles, United States
Probability theory and stochastic processes
General
39
56
1
10.4171/165-1/4
http://www.ems-ph.org/doi/10.4171/165-1/4
Free group factors
Ken
Dykema
Texas A&M University, College Station, USA
Functional analysis
General
57
72
1
10.4171/165-1/5
http://www.ems-ph.org/doi/10.4171/165-1/5
Free convolution
Hari
Bercovici
Indiana University, Bloomington, USA
Operator theory
General
73
93
1
10.4171/165-1/6
http://www.ems-ph.org/doi/10.4171/165-1/6
Easy quantum groups
Moritz
Weber
Universität des Saarlandes, Saarbrücken, Germany
Operator theory
General
95
121
1
10.4171/165-1/7
http://www.ems-ph.org/doi/10.4171/165-1/7
Mathematics and Society
Wolfgang
König
WIAS Berlin and Technical University Berlin, Germany
General
00-XX
Mathematics
Mathematics in the public; mathematics in architecture, biology, climate, encryption, engineering, finance, industry, nature shapes, physics, telecommunication, and voting systems; experimental mathematics, mathematics museums, mathematics for complex data
The ubiquity and importance of mathematics in our complex society is generally not in doubt. However, even a scientifically interested layman would be hard pressed to point out aspects of our society where contemporary mathematical research is essential. Most popular examples are finance, engineering, wheather and industry, but the way mathematics comes into play is widely unknown in the public. And who thinks of application fields like biology, encryption, architecture, or voting systems? This volume comprises a number of success stories of mathematics in our society – important areas being shaped by cutting edge mathematical research. The authors are eminent mathematicians with a high sense for public presentation, addressing scientifically interested laymen as well as professionals in mathematics and its application disciplines.
7
4
2016
978-3-03719-164-4
978-3-03719-664-9
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/164
http://www.ems-ph.org/doi/10.4171/164
The truth, the whole truth and nothing but the truth: The challenges of reporting on mathematics
George
Szpiro
New York, USA
General
1
5
1
10.4171/164-1/1
http://www.ems-ph.org/doi/10.4171/164-1/1
Experimental mathematics in the society of the future
David
Bailey
University of California at Davis, USA
Jonathan
Borwein
The University of Newcastle, Callaghan, Australia
General
Computer-based tools for mathematics are changing how mathematics is researched, taught and communicated to society. Future technology trends point to ever-more powerful tools in the future. Computation in mathematics is thus giving rise to a new mode of mathematical research, where algorithms, datasets and public databases are as significant as the resulting theorems, and even the definition of what constitutes secure mathematical knowledge is seen in a new light.
7
25
1
10.4171/164-1/2
http://www.ems-ph.org/doi/10.4171/164-1/2
What is the impact of interactive mathematical experiments?
Albrecht
Beutelspacher
Mathematikum Giessen, Germany
General
In this article we look at mathematical experiments, in particular those shown in mathematical science centers. Although some mathematicians have the sneaking suspicion that such experiments are far too superficial and do not correspond to proper mathematics, we try to show that in fact mathematical experiments provide an ideal first step for the general public into mathematics.
27
35
1
10.4171/164-1/3
http://www.ems-ph.org/doi/10.4171/164-1/3
Mathematics and finance
Walter
Schachermayer
Universität Wien, Austria
General
This article consists of two parts. The first briefly discusses the history and the basic ideas of option pricing. Based on this background, in the second part we critically analyze the role of academic research in Mathematical Finance relating to the emergence of the 2007–2008 financial crisis.
37
50
1
10.4171/164-1/4
http://www.ems-ph.org/doi/10.4171/164-1/4
Statistics in high dimensions
Aad
van der Vaart
Universiteit Leiden, Netherlands
Wessel
van Wieringen
VU University Medical Center, Amsterdam, Netherlands
Statistics
General
High-dimensional data and models are central to modern statistics. We review some key concepts, some intriguing connections to ideas of the past, and some methods and theoretical results developed in the past decade. We illustrate these results in the context of genomic data from cancer research.
51
70
1
10.4171/164-1/5
http://www.ems-ph.org/doi/10.4171/164-1/5
Filtering theory: Mathematics in engineering, from Gauss to particle filters
Ofer
Zeitouni
Weizmann Institute of Science, Rehovot, Israel
General
The evolution of engineering needs, especially in the areas of estimation and system theory, has triggered the development of mathematical tools which, in turns, have had a profound influence on engineering practice. We describe this interaction through one example, the evolution of filtering theory.
71
80
1
10.4171/164-1/6
http://www.ems-ph.org/doi/10.4171/164-1/6
Mathematical models for population dynamics: Randomness versus determinism
Jean
Bertoin
Universität Zürich, Switzerland
General
Mathematical models are used more and more frequently in Life Sciences. These may be deterministic, or stochastic. We present some classical models for population dynamics and discuss in particular the averaging effect in the setting of large populations, to point at circumstances where randomness prevails nonetheless.
81
99
1
10.4171/164-1/7
http://www.ems-ph.org/doi/10.4171/164-1/7
The quest for laws and structure
Jürg
Fröhlich
ETH Zürich, Switzerland
General
The purpose of this paper is to illustrate, on some concrete examples, the quest of theoretical physicists for new laws of Nature and for appropriate mathematical structures that enables them to formulate and analyze new laws in as simple terms as possible and to derive consequences therefrom. The examples are taken from thermodynamics, atomism and quantum theory.
101
129
1
10.4171/164-1/8
http://www.ems-ph.org/doi/10.4171/164-1/8
Geometry and freeform architecture
Helmut
Pottmann
Technische Universität Wien, Austria
Johannes
Wallner
Technische Universität Graz, Austria
Geometry
General
During the last decade, the geometric aspects of freeform architecture have defined a field of applications which is systematically explored and which conversely serves as inspiration for new mathematical research. This paper discusses topics relevant to the realization of freeform skins by various means (flat and curved panels, straight and curved members, masonry, etc.) and illuminates the interrelations of those questions with theory, in particular discrete differential geometry and discrete conformal geometry.
131
151
1
10.4171/164-1/9
http://www.ems-ph.org/doi/10.4171/164-1/9
Some geometries to describe nature
Christiane
Rousseau
University of Montreal, Canada
Geometry
General
Since ancient times, the development of mathematics has been inspired, at least in part, by the need to provide models in other sciences, and that of describing and understanding the world around us. In this note, we concentrate on the shapes of nature and introduce two related geometries that play an important role in contemporary science. Fractal geometry allows describing a wide range of shapes in nature. In 1973, Harry Blum introduced a new geometry well suited to describe animal morphology.
153
165
1
10.4171/164-1/10
http://www.ems-ph.org/doi/10.4171/164-1/10
Mathematics in industry
Helmut
Neunzert
ITWM, Kaiserslautern, Germany
General
Industrial mathematics has become a fashionable subject in the last three decades. Today, there are many university groups and even research institutes dedicated to industrial and applied mathematics worldwide, proving that mathematics has indeed become a key technology. In this paper we first give a short account of the history of industrial mathematics. Using experiences from the Fraunhofer Institute for Industrial Mathematics (ITWM), we try to characterize the specific problem driven work of industrial mathematicians and take a look at future challenges.
167
183
1
10.4171/164-1/11
http://www.ems-ph.org/doi/10.4171/164-1/11
Mathematics of signal design for communication systems
Holger
Boche
Technische Universität München, Germany
Ezra
Tampubolon
Technische Universität München, Germany
General
Orthogonal transmission schemes constitute the foundations of both our present and future communication standards. One of the major drawback of orthogonal transmission schemes is their high dynamical behaviour, which can be measured by the so called Peakto–Average power value – the ratio between the peak value (i.e. $L^\infty$-norm) and the average power (i.e., $L^2$-norm) of a signal. This undesired behaviour of orthogonal schemes has remarkable negative impacts on the performance, the energy-efficiency, and the maintain cost of the transmission systems. In this work, we give some discussions concerning to the problem of reduction of the high dynamics of an orthogonal transmission scheme. We show that this problem is connected with some mathematical fields, such as functional analysis (Hahn-Banach Theorem and Baire Category), additive combinatorics (Szeméredi Theorem, Green-Tao Theorem on arithmetic progressions in the primes, sparse Szeméredi type Theorems by Conlon and Gowers, and the famous Erd˝os problem on arithmetic progressions), and both trigonometric and non-trigonometric harmonic analysis.
185
220
1
10.4171/164-1/12
http://www.ems-ph.org/doi/10.4171/164-1/12
Cryptology: Methods, applications and challenges
Claus
Diem
Universität Leipzig, Germany
General
Information processing by electronic devices leads to a multitude of security-relevant challenges. With the help of cryptography, many of these challenges can be solved and new applications can be made possible. What methods are hereby used? On which mathematical foundations do they rest? How did the prevailing ideas and methods come about? What are the current developments, what challenges exist and which future challenges can be predicted?
221
250
1
10.4171/164-1/13
http://www.ems-ph.org/doi/10.4171/164-1/13
A mathematical view on voting and power
Werner
Kirsch
FernUniversität Hagen, Germany
General
In this article we describe some concepts, ideas and results from the mathematical theory of voting. We give a mathematical description of voting systems and introduce concepts to measure the power of a voter. We also describe and investigate two-tier voting systems, for example the Council of the European Union. In particular, we prove criteria which give the optimal voting weights in such systems.
251
279
1
10.4171/164-1/14
http://www.ems-ph.org/doi/10.4171/164-1/14
Numerical methods and scientific computing for climate and geosciences
Jörn
Behrens
Universität Hamburg, Germany
General
Studying the climate, weather or other geoscientific phenomena is strongly related to simulation based knowledge gain, since the climate system, for example, is not assessable by laboratory experiments. In these simulations, mathematical models as well as numerical methods play a crucial role in many aspects of the knowledge work-flow. We will describe the general set-up of geoscientific models, and explore some of the applied mathematical methods involved in solving such models. One of the paramount problems of geoscientific simulation applications is the large span of scales that interact.
281
293
1
10.4171/164-1/15
http://www.ems-ph.org/doi/10.4171/164-1/15
Geometry, Analysis and Dynamics on sub-Riemannian Manifolds
Volume II
Davide
Barilari
Université Paris 7 Denis Diderot, Paris, France
Ugo
Boscain
École Polytechnique, Palaiseau, France
Mario
Sigalotti
École Polytechnique, Palaiseau, France
Differential geometry
Partial differential equations
Calculus of variations and optimal control; optimization
Probability theory and stochastic processes
53C17, 35H10, 60H30, 49J15
Differential + Riemannian geometry
Sub-Riemannian geometry, hypoelliptic operators, non-holonomic constraints, optimal control, rough paths
Sub-Riemannian manifolds model media with constrained dynamics: motion at any point is only allowed along a limited set of directions, which are prescribed by the physical problem. From the theoretical point of view, sub-Riemannian geometry is the geometry underlying the theory of hypoelliptic operators and degenerate diffusions on manifolds. In the last twenty years, sub-Riemannian geometry has emerged as an independent research domain, with extremely rich motivations and ramifications in several parts of pure and applied mathematics, such as geometric analysis, geometric measure theory, stochastic calculus and evolution equations together with applications in mechanics, optimal control and biology. The aim of the lectures collected here is to present sub-Riemannian structures for the use of both researchers and graduate students.
10
25
2016
978-3-03719-163-7
978-3-03719-663-2
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/163
http://www.ems-ph.org/doi/10.4171/163
EMS Series of Lectures in Mathematics
2523-5176
2523-5184
Introduction to geodesics in sub-Riemannian geometry
Andrei
Agrachev
SISSA, Trieste, Italy
Davide
Barilari
Université Paris 7, Denis Diderot, Paris, France
Ugo
Boscain
Ecole Polytechnique, Palaiseau, France
General
1
83
1
10.4171/163-1/1
http://www.ems-ph.org/doi/10.4171/163-1/1
Geometry of subelliptic diffusions
Anton
Thalmaier
Université du Luxembourg, Luxembourg
General
85
169
1
10.4171/163-1/2
http://www.ems-ph.org/doi/10.4171/163-1/2
Geometric foundations of rough paths
Peter
Friz
Technische Universität Berlin, Germany
Paul
Gassiat
Université Paris-Dauphine, Paris, France
General
171
210
1
10.4171/163-1/3
http://www.ems-ph.org/doi/10.4171/163-1/3
Sobolev and bounded variation functions on metric measure spaces
Luigi
Ambrosio
Scuola Normale Superiore, Pisa, Italy
Roberta
Ghezzi
Université de Bourgogne, Dijon, France
General
211
273
1
10.4171/163-1/4
http://www.ems-ph.org/doi/10.4171/163-1/4
Singularities of vector distributions
Michail
Zhitomirskii
Technion - Israel Institute of Technology, Haifa, Israel
General
275
295
1
10.4171/163-1/5
http://www.ems-ph.org/doi/10.4171/163-1/5
Geometry, Analysis and Dynamics on sub-Riemannian Manifolds
Volume I
Davide
Barilari
Université Paris 7 Denis Diderot, Paris, France
Ugo
Boscain
École Polytechnique, Palaiseau, France
Mario
Sigalotti
École Polytechnique, Palaiseau, France
Differential geometry
Partial differential equations
Calculus of variations and optimal control; optimization
Probability theory and stochastic processes
53C17, 35H10, 60H30, 49J15
Differential + Riemannian geometry
Sub-Riemannian geometry, hypoelliptic operators, non-holonomic constraints, optimal control, rough paths
Sub-Riemannian manifolds model media with constrained dynamics: motion at any point is only allowed along a limited set of directions, which are prescribed by the physical problem. From the theoretical point of view, sub-Riemannian geometry is the geometry underlying the theory of hypoelliptic operators and degenerate diffusions on manifolds. In the last twenty years, sub-Riemannian geometry has emerged as an independent research domain, with extremely rich motivations and ramifications in several parts of pure and applied mathematics, such as geometric analysis, geometric measure theory, stochastic calculus and evolution equations together with applications in mechanics, optimal control and biology. The aim of the lectures collected here is to present sub-Riemannian structures for the use of both researchers and graduate students.
6
30
2016
978-3-03719-162-0
978-3-03719-662-5
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/162
http://www.ems-ph.org/doi/10.4171/162
EMS Series of Lectures in Mathematics
2523-5176
2523-5184
Some topics of geometric measure theory in Carnot groups
Francesco
Serra Cassano
Università di Trento, Italy
Differential geometry
General
1
121
1
10.4171/162-1/1
http://www.ems-ph.org/doi/10.4171/162-1/1
Hypoelliptic operators and some aspects of analysis and geometry of sub-Riemannian spaces
Nicola
Garofalo
Università di Padova, Italy
Differential geometry
General
123
257
1
10.4171/162-1/2
http://www.ems-ph.org/doi/10.4171/162-1/2
Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations
Fabrice
Baudoin
Purdue University, West Lafayette, USA
Differential geometry
General
259
321
1
10.4171/162-1/3
http://www.ems-ph.org/doi/10.4171/162-1/3
Handbook of Teichmüller Theory, Volume VI
Athanase
Papadopoulos
Université de Strasbourg, France
Functions of a complex variable
Primary 30-00, 32-00, 57-00, 32G13, 32G15, 30F60. Secondary 11F06, 11F75, 14D20, 11G32, 14C05, 14H15, 14H30, 14H15, 14H60, 14H55, 14J60, 18A22, 20F14, 20F28, 20F38, 20F65, 20F67, 20H10, 22E46, 30-03, 30C62, 30F20, 30F25, 30F10, 30F15, 30F30, 30F35, 30F40, 30F45, 32-03, 32S30, 32G13, 32G15, 37-99, 53A35, 53B35, 53C35, 53C50, 53C80, 53D55, 53Z05, 57M07, 57M20, 57M27, 57M50, 57M60, 57N16
Functional analysis
This volume is the sixth in a series dedicated to Teichmüller theory in a broad sense, including various moduli and deformation spaces, and the study of mapping class groups. It is divided into five parts: Part A: The metric and the analytic theory. Part B: The group theory. Part C: Representation theory and generalized structures. Part D: The Grothendieck–Teichmüller theory. Part D: Sources. The topics surveyed include Grothendieck’s construction of the analytic structure of Teichmüller space, identities on the geodesic length spectrum of hyperbolic surfaces (including Mirzakhani’s application to the computation of Weil–Petersson volumes), moduli spaces of configurations spaces, the Teichmüller tower with the action of the Galois group on dessins d’enfants, and several others actions related to surfaces. The last part contains three papers by Teichmüller, translated into English with mathematical commentaries, and a document that contains H. Grötzsch’s comments on Teichmüller’s famous paper Extremale quasikonforme Abbildungen und quadratische Differentiale. The handbook is addressed to researchers and to graduate students.
5
31
2016
978-3-03719-161-3
978-3-03719-661-8
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/161
http://www.ems-ph.org/doi/10.4171/161
IRMA Lectures in Mathematics and Theoretical Physics
2523-5133
2523-5141
27
Introduction to Teichmüller theory, old and new, VI
Athanase
Papadopoulos
Université de Strasbourg, France
General
1
29
1
10.4171/161-1/1
http://www.ems-ph.org/doi/10.4171/161-1/1
Alexander Grothendieck
Valentin
Poénaru
Université Paris-Sud 11, Orsay, France
History and biography
General
31
32
1
10.4171/161-1/2
http://www.ems-ph.org/doi/10.4171/161-1/2
On Grothendieck’s construction of Teichmüller space
Norbert
A’Campo
Universität Basel, Switzerland
Lizhen
Ji
University of Michigan, Ann Arbor, USA
Athanase
Papadopoulos
Université de Strasbourg, France
General
35
69
1
10.4171/161-1/3
http://www.ems-ph.org/doi/10.4171/161-1/3
Null-set compactifications of Teichmüller spaces
Vincent
Alberge
Université de Strasbourg, France
Hideki
Miyachi
Osaka University, Japan
Ken’ichi
Ohshika
Osaka University Graduate School of Science, Japan
General
71
94
1
10.4171/161-1/4
http://www.ems-ph.org/doi/10.4171/161-1/4
Mirzakhani’s recursion formula on Weil–Petersson volume and applications
Yi
Huang
University of Melbourne, Melbourne, Victoria, Australia
General
95
127
1
10.4171/161-1/5
http://www.ems-ph.org/doi/10.4171/161-1/5
Rigidity phenomena in the mapping class group
Javier
Aramayona
Universidad Autónoma de Madrid, Spain
Juan
Souto
Université de Rennes 1, France
General
131
165
1
10.4171/161-1/6
http://www.ems-ph.org/doi/10.4171/161-1/6
Harmonic volume and its applications
Yuuki
Tadokoro
Kisarazu National College of Technology, Chiba, Japan
General
167
193
1
10.4171/161-1/7
http://www.ems-ph.org/doi/10.4171/161-1/7
Torus bundles and 2-forms on the universal family of Riemann surfaces
Robin
de Jong
Universiteit Leiden, Netherlands
General
195
227
1
10.4171/161-1/8
http://www.ems-ph.org/doi/10.4171/161-1/8
Cubic Differentials in the Differential Geometry of Surfaces
John
Loftin
Rutgers University, Newark, USA
Ian
McIntosh
University of York, UK
General
231
274
1
10.4171/161-1/9
http://www.ems-ph.org/doi/10.4171/161-1/9
Two-generator groups acting on the complex hyperbolic plane
Pierre
Will
Université de Grenoble I, Saint-Martin d'Hères, France
Functions of a complex variable
General
275
334
1
10.4171/161-1/10
http://www.ems-ph.org/doi/10.4171/161-1/10
Configuration spaces of planar linkages
Alexey
Sossinsky
Independent University of Moscow, Russian Federation
Functions of a complex variable
General
335
373
1
10.4171/161-1/11
http://www.ems-ph.org/doi/10.4171/161-1/11
Quasiconformal mappings on the Heisenberg group: An overview
Ioannis
Platis
University of Crete, Heraklion, Greece
Functions of a complex variable
General
375
393
1
10.4171/161-1/12
http://www.ems-ph.org/doi/10.4171/161-1/12
Actions of the absolute Galois group
Norbert
A’Campo
Universität Basel, Switzerland
Lizhen
Ji
University of Michigan, Ann Arbor, USA
Athanase
Papadopoulos
Université de Strasbourg, France
Functions of a complex variable
General
397
435
1
10.4171/161-1/13
http://www.ems-ph.org/doi/10.4171/161-1/13
A primer on dessins
Pierre
Guillot
Université de Strasbourg, France
Functions of a complex variable
General
437
466
1
10.4171/161-1/14
http://www.ems-ph.org/doi/10.4171/161-1/14
Hypergeometric Galois Actions
A. Muhammed
Uludağ
Galatasaray University, Istanbul, Turkey
İsmail
Sağlam
Galatasaray University, Istanbul, Turkey
Functions of a complex variable
General
467
500
1
10.4171/161-1/15
http://www.ems-ph.org/doi/10.4171/161-1/15
A panaroma of the fundamental group of the modular orbifold
A. Muhammed
Uludağ
Galatasaray University, Istanbul, Turkey
Ayberk
Zeytin
Galatasaray University, Istanbul, Turkey
Functions of a complex variable
General
501
519
1
10.4171/161-1/16
http://www.ems-ph.org/doi/10.4171/161-1/16
On Grothendieck’s tame topology
Norbert
A’Campo
Universität Basel, Switzerland
Lizhen
Ji
University of Michigan, Ann Arbor, USA
Athanase
Papadopoulos
Université de Strasbourg, France
Functions of a complex variable
General
521
533
1
10.4171/161-1/17
http://www.ems-ph.org/doi/10.4171/161-1/17
Some historical commentaries on Teichmüller’s paper Extremale quasikonforme Abbildungen und quadratische Differentiale
Reiner
Kühnau
Martin-Luther-Universität Halle-Wittenberg, Germany
Functions of a complex variable
General
537
546
1
10.4171/161-1/18
http://www.ems-ph.org/doi/10.4171/161-1/18
Complete solution of an extremal problem of the quasiconformal mapping
Oswald
Teichmüller
Berlin, Germany
Functions of a complex variable
General
547
560
1
10.4171/161-1/19
http://www.ems-ph.org/doi/10.4171/161-1/19
A Commentary on Teichmüller’s paper Vollständige Lösung einer Extremalaufgabe der quasikonformen Abbildung
Vincent
Alberge
Université de Strasbourg, France
Athanase
Papadopoulos
Université de Strasbourg, France
Functions of a complex variable
General
561
567
1
10.4171/161-1/20
http://www.ems-ph.org/doi/10.4171/161-1/20
On extremal problems of conformal geometry
Oswald
Teichmüller
Berlin, Germany
Functions of a complex variable
General
569
596
1
10.4171/161-1/21
http://www.ems-ph.org/doi/10.4171/161-1/21
A Commentary on Teichmüller’s paper Über Extremalprobleme der konformen Geometrie
Norbert
A’Campo
Universität Basel, Switzerland
Athanase
Papadopoulos
Université de Strasbourg, France
Functions of a complex variable
General
597
603
1
10.4171/161-1/22
http://www.ems-ph.org/doi/10.4171/161-1/22
A displacement theorem of quasiconformal mapping
Oswald
Teichmüller
Berlin, Germany
Functions of a complex variable
General
605
612
1
10.4171/161-1/23
http://www.ems-ph.org/doi/10.4171/161-1/23
A Commentary on Teichmüller’s paper Ein Verschiebungssatz der quasikonformen Abbildung
Vincent
Alberge
Université de Strasbourg, France
Functions of a complex variable
General
613
629
1
10.4171/161-1/24
http://www.ems-ph.org/doi/10.4171/161-1/24
Handbook of Teichmüller Theory, Volume V
Athanase
Papadopoulos
Université de Strasbourg, France
Functions of a complex variable
Primary 30-00, 32-00, 57-00, 32G13, 32G15, 30F60; Secondary 11F06, 11F75, 14D20, 14H15, 14H60, 14H55, 14J60, 20F14, 20F28, 20F38, 20F65, 20F67, 20H10, 22E46, 30-03, 30C62, 30F20, 30F25, 30F10, 30F15, 30F30, 30F35, 30F40, 30F45, 32-03, 32S30, 37-99, 53A35, 53B35, 53C35, 53C50, 53C80, 53D55, 53Z05, 57M07, 57M20, 57M27, 57M50, 57M60, 57N16
Functional analysis
This volume is the fifth in a series dedicated to Teichmüller theory in a broad sense, including the study of various deformation spaces and of mapping class group actions. It is divided into four parts: Part A: The metric and the analytic theory Part B: The group theory Part C: Representation theory and generalized structures Part D: Sources The topics that are covered include identities for the hyperbolic geodesic length spectrum, Thurston's metric, the cohomology of moduli space and mapping class groups, the Johnson homomorphisms, Higgs bundles, dynamics on character varieties, and there are many others. Besides surveying important parts of the theory, several chapters contain conjectures and open problems. The last part contains two fundamental papers by Teichmüller, translated into English and accompanied by mathematical commentaries. The chapters, like those of the other volumes in this collection, are written by experts who have a broad view on the subject. They have an expository character (which fits with the original purpose of this handbook), but some of them also contain original and new material. The Handbook is addressed to researchers and to graduate students.
1
11
2016
978-3-03719-160-6
978-3-03719-660-1
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/160
http://www.ems-ph.org/doi/10.4171/160
IRMA Lectures in Mathematics and Theoretical Physics
2523-5133
2523-5141
26
Introduction to Teichmüller theory, old and new, V
Athanase
Papadopoulos
Université de Strasbourg, France
General
1
16
1
10.4171/160-1/1
http://www.ems-ph.org/doi/10.4171/160-1/1
Identities on hyperbolic manifolds
Martin
Bridgeman
Boston College, Chestnut Hill, USA
Ser Peow
Tan
National University of Singapore, Singapore
Hyperbolic manifolds, identities, orthogeodesic, ortholength, orthospec- trum, simple geodesics, geodesic flow
Manifolds and cell complexes
Group theory and generalizations
Functions of a complex variable
General
In this survey, we discuss four classes of identities due principally to Basmajian, McShane, Bridgeman-Kahn and Luo-Tan on hyperbolic manifolds and provide a unified approach for proving them. We also elucidate on the connections between the various identities.
19
53
1
10.4171/160-1/2
http://www.ems-ph.org/doi/10.4171/160-1/2
Problems on the Thurston metric
Weixu
Su
Fudan University, Shanghai, China
Teichmüller space, Thurston metric
Several complex variables and analytic spaces
Functions of a complex variable
General
We present a list of problems related to the study of Thurston metric on Teichmüller space. The problems originate in discussions by the participants at the 2012 AIM workshop “Lipschitz metric on Teichmüller space”. Some of the problems were worked on by small groups, and they remain open. Some others were suggested by various participants at the closing problem session of the conference or added after the conference. We have updated the list in order to include it in the present Handbook.
55
72
1
10.4171/160-1/3
http://www.ems-ph.org/doi/10.4171/160-1/3
Meyer functions and the signature of fibered 4-manifolds
Yusuke
Kuno
Tsuda College, Tokyo, Japan
The signature cocycle, Meyer function, local signature
Algebraic geometry
Group theory and generalizations
Several complex variables and analytic spaces
Manifolds and cell complexes
We give a survey on Meyer functions, with emphasis on their application to the signature of fibered 4-manifolds.
75
96
1
10.4171/160-1/4
http://www.ems-ph.org/doi/10.4171/160-1/4
The Goldman–Turaev Lie bialgebra and the Johnson homomorphisms
Nariya
Kawazumi
University of Tokyo, Japan
Yusuke
Kuno
Tsuda College, Tokyo, Japan
Mapping class group, Johnson homomorphism, Goldman-Turaev Lie bial- gebra, Dehn twist
Group theory and generalizations
Several complex variables and analytic spaces
Manifolds and cell complexes
General
We survey a geometric approach to the Johnson homomorphisms using the Goldman–Turaev Lie bialgebra.
97
165
1
10.4171/160-1/5
http://www.ems-ph.org/doi/10.4171/160-1/5
A survey of the Johnson homomorphisms of the automorphism groups of free groups and related topics
Takao
Satoh
Tokyo University of Science, Japan
Automrophism groups of free groups, IA-automorphism groups, Johnson homomorphisms, Magnus representations
Group theory and generalizations
General
This is a survey on the Johnson homomorphisms of the automorphism groups of free groups. We exposit some well known facts and recent developments for the Johnson homomorphisms and its related topics.
167
209
1
10.4171/160-1/6
http://www.ems-ph.org/doi/10.4171/160-1/6
Geometry and dynamics on character varieties
Inkang
Kim
KIAS, Seoul, South Korea
Character variety, rigidity and flexibility, bounded cohomology
Topological groups, Lie groups
Differential geometry
Manifolds and cell complexes
General
We survey recent progress on character varieties. We give a general definition of a character variety, yet focus on rigidity and flexibility issues of a given representation. On the other hand, we study dynamical aspects on the character variety, the relation to bounded cohomology theory and several topological issues using different branches of mathematics. We try to give as many familiar examples as possible to motivate the readers.
213
235
1
10.4171/160-1/7
http://www.ems-ph.org/doi/10.4171/160-1/7
Compactifications and reduction theory of geometrically finite locally symmetric spaces
Lizhen
Ji
University of Michigan, Ann Arbor, USA
Satake compactification, Anosov subgroup, reduction theory, coarse fundamental domain, Borel-Serre compactification, geometrically finite space, locally symmetric space
Topological groups, Lie groups
Algebraic geometry
General
There are two closely related classes of groups arising from Fuchsian groups and their actions on the hyperbolic plane $\H^2$: discrete subgroups of semisimple Lie groups acting on symmetric spaces, and mapping class groups and their subgroups acting on Teichmüller space. Convex cocompact Fuchsian groups have been generalized to Anosov subgroups of semisimple Lie groups of higher rank, which have played an important role in higher Teichmüller theory. They have also been generalized to convex cocompact subgroups of mapping class groups. Due to the fact that Teichmüller space shares some similarities with symmetric spaces of rank one, the analogy between convex cocompact Fuchsian groups and convex cocompact subgroups of mapping class groups is rather complete. But less is known about actions of Anosov subgroups on symmetric spaces of higher rank. In this chapter, we discuss three conjectures on compactifications and coarse fundamental domains for locally symmetric spaces associated with Anosov subgroups of noncompact semisimple Lie groups, and describe results, motivations, and evidence for these conjectures. One conjecture deals with the existence of maximal Satake compactifications of Anosov locally symmetric spaces, and other two are concerned with the characterization of Anosov subgroups. By comparing these locally symmetric spaces of infinite volume with symmetric spaces and locally symmetric spaces of finite volume, and by examining applications of the maximal Satake compactification of symmetric spaces and the Borel-Serre compactification of locally symmetric spaces of finite volume, we conclude that the conjectural maximal Satake compactifications of Anosov locally symmetric spaces arising from the maximal Satake compactification of symmetric spaces are the natural compactifications. In general, there is more than one maximal Satake compactification for Anosov locally symmetric spaces. To explain this non-uniqueness, we develop a reduction theory for Anosov subgroups by introducing the crucial notion of anti-Siegel sets. We also explain how geometric boundaries of the maximal Satake compactifications are related to problems in the spectral theory of Laplace operators of the Riemannian manifolds under consideration, and conclude with some questions on the Martin compactification of Anosov locally symmetric spaces, which is related to positive harmonic functions.
237
305
1
10.4171/160-1/8
http://www.ems-ph.org/doi/10.4171/160-1/8
Representations of fundamental groups of 2-manifolds
Lisa
Jeffrey
University of Toronto, Canada
Symplectic geometry, representations, fundamental groups
Global analysis, analysis on manifolds
Partial differential equations
General
This chapter aims to provide a survey on the subject of representations of fundamental groups of 2-manifolds, or in other guises flat connections on orientable 2-manifolds or moduli spaces parametrizing holomorphic vector bundles on Riemann surfaces. It emphasizes the relationships between the different descriptions of these spaces.
307
317
1
10.4171/160-1/9
http://www.ems-ph.org/doi/10.4171/160-1/9
Extremal quasiconformal mappings and quadratic differentials
Oswald
Teichmüller
Berlin, Germany
General
321
483
1
10.4171/160-1/10
http://www.ems-ph.org/doi/10.4171/160-1/10
A commentary on Teichmüller’s paper Extremale quasikonforme Abbildungen und quadratische Differentiale
Vincent
Alberge
Université de Strasbourg, France
Athanase
Papadopoulos
Université de Strasbourg, France
Weixu
Su
Fudan University, Shanghai, China
General
We provide a mathematical commentary on Teichmüller’s paper Extremale quasikonforme Abbildungen und quadratische Differentiale (Extremal quasiconformal mappings of closed oriented Riemann surfaces), Abh. Preuss. Akad. Wiss., Math.- Naturw. Kl. 1940, No.22, 1–197 (1940). The paper is quoted in several works, although it was read by very few people. Some of the results it contains were re- discovered later on and published without any reference to Teichmüller. In this commentary, we highlight the main results and the main ideas contained in that paper and we describe some of the important developments they gave rise to.
485
531
1
10.4171/160-1/11
http://www.ems-ph.org/doi/10.4171/160-1/11
Determination of extremal quasiconformal mappings of closed oriented Riemann surfaces
Oswald
Teichmüller
Berlin, Germany
General
533
567
1
10.4171/160-1/12
http://www.ems-ph.org/doi/10.4171/160-1/12
A commentary on Teichmüller’s paper Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen
Annette
A’Campo-Neuen
Universität Basel, Switzerland
Norbert
A’Campo
Universität Basel, Switzerland
Vincent
Alberge
Université de Strasbourg, France
Athanase
Papadopoulos
Université de Strasbourg, France
General
This is a mathematical commentary on Teichmüller’s paper Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen (Determination of extremal quasiconformal maps of closed oriented Riemann surfaces) [24], (1943). This paper is among the last (and may be the last one) that Teichmüller wrote on the theory of moduli. It contains the proof of the so-called Teichmüller existence theorem for a closed surface of genus $g ≥ 2$. For this proof, the author defines a mapping between a space of equivalence classes of marked Riemann surfaces (the Teichmüller space) and a space of equivalence classes of certain Fuchsian groups (the so-called Fricke space). After that, he defines a map between the latter and the Euclidean space of dimension $6g−6$. Using Brouwer’s theorem of invariance of domain, he shows that this map is a home- omorphism. This involves in particular a careful definition of the topologies of Fricke space, the computation of its dimension, and comparison results between hyperbolic distance and quasiconformal dilatation. The use of the invariance of domain theorem is in the spirit of Poincar ́e and Klein’s use of the so-called “continuity principle” in their attempts to prove the uniformization theorem.
569
580
1
10.4171/160-1/13
http://www.ems-ph.org/doi/10.4171/160-1/13
Measure and Integration
Dietmar
Salamon
ETH Zürich, Switzerland
Measure and integration
Primary: 28-01; Secondary: 28C05, 28C10, 28C15, 35J05, 43A05, 44A35, 46B22, 46C05, 46E27, 46E30
Real analysis
sigma-Algebra, Lebesgue monotone convergence, Caratheodory criterion, Lebesgue measure, Borel measure, Dieudonné’s measure, Riesz Representation Theorem, Alexandrov Double Arrow Space, Sorgenfrey Line, separability, Cauchy–Schwarz inequality, Jensen’s inequality, Egoroff’s theorem, Hardy’s inequality, absolutely continuous measure, truly continuous measure, singular measure, signed measure, Radon–Nikodym Theorem, Lebesgue Decomposition Theorem, Hahn Decomposition Theorem, Jordan Decomposition Theorem, Hardy–Littlewood maximal inequality, Vitali’s Covering Lemma, Lebesgue point, Lebesgue Differentiation Theorem, Banach-Zarecki Theorem, Vitali–Caratheodory Theorem, Cantor function, product sigma-algebra, Fubini’s Theorem, convolution, Young’s inequality, mollifier, Marcinciewicz interpolation, Poisson identity, Green’s formula, Calderon–Zygmund inequality, Haar measure, modular character
The book is intended as a companion to a one semester introductory lecture course on measure and integration. After an introduction to abstract measure theory it proceeds to the construction of the Lebesgue measure and of Borel measures on locally compact Hausdorff spaces, $L^p$ spaces and their dual spaces and elementary Hilbert space theory. Special features include the formulation of the Riesz Representation Theorem in terms of both inner and outer regularity, the proofs of the Marcinkiewicz Interpolation Theorem and the Calderon–Zygmund inequality as applications of Fubini’s theorem and Lebesgue differentiation, the treatment of the generalized Radon–Nikodym theorem due to Fremlin, and the existence proof for Haar measures. Three appendices deal with Urysohn’s Lemma, product topologies, and the inverse function theorem. The book assumes familiarity with first year analysis and linear algebra. It is suitable for second year undergraduate students of mathematics or anyone desiring an introduction to the concepts of measure and integration.
3
29
2016
978-3-03719-159-0
978-3-03719-659-5
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/159
http://www.ems-ph.org/doi/10.4171/159
EMS Textbooks in Mathematics
Metric Measure Geometry
Gromov’s Theory of Convergence and Concentration of Metrics and Measures
Takashi
Shioya
Tohoku University, Sendai, Japan
Differential geometry
Measure and integration
Functions of a complex variable
Probability theory and stochastic processes
Primary: 53C23; Secondary: 28A33, 30Lxx, 35P15, 53C20, 54Exx, 58C40, 58J50, 60B10
Differential + Riemannian geometry
Probability + statistics
Metric measure space, concentration of measure phenomenon, observable distance, pyramid, convergence of spaces, curvature-dimension condition, Laplacian, dissipation
This book studies a new theory of metric geometry on metric measure spaces, originally developed by M. Gromov in his book “Metric Structures for Riemannian and Non-Riemannian Spaces” and based on the idea of the concentration of measure phenomenon due to Lévy and Milman. A central theme in this text is the study of the observable distance between metric measure spaces, defined by the difference between 1-Lipschitz functions on one space and those on the other. The topology on the set of metric measure spaces induced by the observable distance function is weaker than the measured Gromov–Hausdorff topology and allows to investigate a sequence of Riemannian manifolds with unbounded dimensions. One of the main parts of this presentation is the discussion of a natural compactification of the completion of the space of metric measure spaces. The stability of the curvature-dimension condition is also discussed. This book makes advanced material accessible to researchers and graduate students interested in metric measure spaces.
1
5
2016
978-3-03719-158-3
978-3-03719-658-8
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/158
http://www.ems-ph.org/doi/10.4171/158
IRMA Lectures in Mathematics and Theoretical Physics
2523-5133
2523-5141
25
Absolute Arithmetic and $\mathbb F_1$-Geometry
Koen
Thas
University of Gent, Belgium
Combinatorics
Number theory
Commutative rings and algebras
Algebraic geometry
Primary: 05E18, 11M26, 13F35, 13K05, 14A15, 14A20, 14A22, 14G15, 14G40, 14H10, 18A05, 19E08, 20B25, 20G05, 20G35, 20M25, 51E24; Secondary: 05E05, 06B10, 11G20, 11G25, 11R18, 11T55, 13A35, 13C60, 14C40, 14F05, 14L15, 14M25, 14M26, 14P10, 15B48, 16G20, 16Y60, 18D50, 18F20, 20E42, 20F36, 20M14, 20M32, 20N20, 51B25, 55N30, 55P42, 55Q45
Combinatorics + graph theory
The field with one element, $\mathbb F_1$-geometry, combinatorial $\mathbb F_1$-geometry, non-additive category, Deitmar scheme, graph, monoid, motive, zeta function, automorphism group, blueprint, Euler characteristic, K-theory, Grassmannian, Witt ring, noncommutative geometry, Witt vector, total positivity, moduli space of curves, operad, torificiation, Absolute Arithmetic, counting function, Weil conjectures, Riemann Hypothesis
It has been known for some time that geometries over finite fields, their automorphism groups and certain counting formulae involving these geometries have interesting guises when one lets the size of the field go to 1. On the other hand, the nonexistent field with one element, $\mathbb F_1$, presents itself as a ghost candidate for an absolute basis in Algebraic Geometry to perform the Deninger–Manin program, which aims at solving the classical Riemann Hypothesis. This book, which is the first of its kind in the $\mathbb F_1$-world, covers several areas in $\mathbb F_1$-theory, and is divided into four main parts – Combinatorial Theory, Homological Algebra, Algebraic Geometry and Absolute Arithmetic. Topics treated include the combinatorial theory and geometry behind $\mathbb F_1$, categorical foundations, the blend of different scheme theories over $\mathbb F_1$ which are presently available, motives and zeta functions, the Habiro topology, Witt vectors and total positivity, moduli operads, and at the end, even some arithmetic. Each chapter is carefully written by experts, and besides elaborating on known results, brand new results, open problems and conjectures are also met along the way. The diversity of the contents, together with the mystery surrounding the field with one element, should attract any mathematician, regardless of speciality.
7
25
2016
978-3-03719-157-6
978-3-03719-657-1
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/157
http://www.ems-ph.org/doi/10.4171/157
The Weyl functor. Introduction to Absolute Arithmetic
Koen
Thas
Universiteit Gent, Belgium
Combinatorics
Number theory
Commutative rings and algebras
Algebraic geometry
3
36
1
10.4171/157-1/1
http://www.ems-ph.org/doi/10.4171/157-1/1
Belian categories
Anton
Deitmar
Universität Tübingen, Germany
Combinatorics
Number theory
Commutative rings and algebras
Algebraic geometry
39
80
1
10.4171/157-1/2
http://www.ems-ph.org/doi/10.4171/157-1/2
The combinatorial-motivic nature of $\mathbb F_1$-schemes
Koen
Thas
Universiteit Gent, Belgium
Combinatorics
Number theory
Commutative rings and algebras
Algebraic geometry
83
159
1
10.4171/157-1/3
http://www.ems-ph.org/doi/10.4171/157-1/3
A blueprinted view on $\mathbb F_1$-geometry
Oliver
Lorscheid
Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, Brazil
Combinatorics
Number theory
Commutative rings and algebras
Algebraic geometry
161
219
1
10.4171/157-1/4
http://www.ems-ph.org/doi/10.4171/157-1/4
Absolute geometry and the Habiro topology
Lieven
Le Bruyn
University of Antwerp, Belgium
Combinatorics
Number theory
Commutative rings and algebras
Algebraic geometry
221
271
1
10.4171/157-1/5
http://www.ems-ph.org/doi/10.4171/157-1/5
Witt vectors, semirings, and total positivity
James
Borger
Australian National University, Canberra, Australia
Combinatorics
Number theory
Commutative rings and algebras
Algebraic geometry
273
329
1
10.4171/157-1/6
http://www.ems-ph.org/doi/10.4171/157-1/6
Moduli operad over $\mathbb F_1$
Yuri
Manin
Universität Bonn, Germany
Matilde
Marcolli
California Institute of Technology, Pasadena, United States
Combinatorics
Number theory
Commutative rings and algebras
Algebraic geometry
331
361
1
10.4171/157-1/7
http://www.ems-ph.org/doi/10.4171/157-1/7
A taste of Weil theory in characteristic one
Koen
Thas
Universiteit Gent, Belgium
Combinatorics
Number theory
Commutative rings and algebras
Algebraic geometry
365
386
1
10.4171/157-1/8
http://www.ems-ph.org/doi/10.4171/157-1/8
Higher-Dimensional Generalized Manifolds: Surgery and Constructions
Alberto
Cavicchioli
Università degli Studi di Modena e Reggio Emilia, Italy
Friedrich
Hegenbarth
Università degli Studi di Milano, Italy
Dušan
Repovš
University of Ljubljana, Slovenia
Manifolds and cell complexes
Category theory; homological algebra
$K$-theory
Primary: 57P05, 57P10, 57P99, 57R65, 57R67; Secondary: 18F15, 19J25, 57N15, 57N60, 57N65
Topology
Homology manifold, Poincaré duality, degree 1 normal map, boundedly controlled surgey, surgery spectrum, assembly map, Quinn index, Euclidean neighborhood retract, cell-like resolution, disjoint disks property, manifold recognition problem
Generalized manifolds are a most fascinating subject to study. They were introduced in the 1930s, when topologists tried to detect topological manifolds among more general spaces (this is nowadays called the manifold recognition problem). As such, generalized manifolds have served to understand the nature of genuine manifolds. However, it soon became more important to study the category of generalized manifolds itself. A breakthrough was made in the 1990s, when several topologists discovered a systematic way of constructing higher-dimensional generalized manifolds, based on advanced surgery techniques. In fact, the development of controlled surgery theory and the study of generalized manifolds developed in parallel. In this process, earlier studies of geometric surgery turned out to be very helpful. Generalized manifolds will continue to be an attractive subject to study, for there remain several unsolved fundamental problems. Moreover, they hold promise for new research, e.g. for finding appropriate structures on these spaces which could bring to light geometric (or even analytic) aspects of higher-dimensional generalized manifolds. This is the first book to systematically collect the most important material on higher-dimensional generalized manifolds and controlled surgery. It is self-contained and its extensive list of references reflects the historic development. The book is based on our graduate courses and seminars, as well as our talks given at various meetings, and is suitable for advanced graduate students and researchers in algebraic and geometric topology.
5
31
2016
978-3-03719-156-9
978-3-03719-656-4
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/156
http://www.ems-ph.org/doi/10.4171/156
EMS Series of Lectures in Mathematics
2523-5176
2523-5184
Tempered Homogeneous Function Spaces
Hans
Triebel
University of Jena, Germany
Functional analysis
Fourier analysis
Functional analysis
If one tries to transfer assertions for the inhomogeneous spaces $A^s_{p,q} (\mathbb R^n)$, $A \in \{B,F \}$, appropriately to their homogeneous counterparts ${\overset {\, \ast}{A}}{}^s_{p,q} (\mathbb R^n)$ within the framework of the dual pairing $\big( S(\mathbb R^n), S'(\mathbb R^n) \big)$ then it is hard to make a mistake as long as the parameters $p,q,s$ are restricted by $0 < p,q \le \infty$ and, in particular, $n(\frac {1}{p} – 1) < s < \frac {n}{p}$. It is the main aim of these notes to say what this means. This book is addressed to graduate students and mathematicians having a working knowledge of basic elements of the theory of function spaces, especially of type $B^s_{p,q}$ and $F^s_{p,q}$.
9
30
2015
978-3-03719-155-2
978-3-03719-655-7
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/155
http://www.ems-ph.org/doi/10.4171/155
EMS Series of Lectures in Mathematics
2523-5176
2523-5184
3-Manifold Groups
Matthias
Aschenbrenner
University of California Los Angeles, USA
Stefan
Friedl
Universität Regensburg, Germany
Henry
Wilton
University of Cambridge, UK
Manifolds and cell complexes
Group theory and generalizations
Primary: 57M05, 57M27; Secondary: 20E26
Groups + group theory
3-manifolds, fundamental groups, Geometrization theorem, hyperbolic 3-manifolds, virtually special groups
The field of 3-manifold topology has made great strides forward since 1982, when Thurston articulated his influential list of questions. Primary among these is Perelman's proof of the Geometrization Conjecture, but other highlights include the Tameness Theorem of Agol and Calegari–Gabai, the Surface Subgroup Theorem of Kahn–Markovic, the work of Wise and others on special cube complexes, and finally Agol's proof of the Virtual Haken Conjecture. This book summarizes all these developments and provides an exhaustive account of the current state of the art of 3-manifold topology, especially focussing on the consequences for fundamental groups of 3-manifolds. As the first book on 3-manifold topology that incorporates the exciting progress of the last two decades, it will be an invaluable resource for researchers in the field who need a reference for these developments. It also gives a fast-paced introduction to this material – although some familiarity with the fundamental group is recommended, little other previous knowledge is assumed, and the book is accessible to graduate students. The book closes with an extensive list of open questions, which will also be of interest to graduate students and established researchers alike.
8
20
2015
978-3-03719-154-5
978-3-03719-654-0
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/154
http://www.ems-ph.org/doi/10.4171/154
EMS Series of Lectures in Mathematics
2523-5176
2523-5184
Free Loop Spaces in Geometry and Topology
Including the monograph Symplectic cohomology and Viterbo’s theorem by Mohammed Abouzaid
Janko
Latschev
Universität Hamburg, Germany
Alexandru
Oancea
Sorbonne Universités, Paris, France
Differential geometry
Commutative rings and algebras
Associative rings and algebras
Algebraic topology
Primary: 53D40, 53D12, 53D25, 53D35, 53D37, 55P35, 55P50, 55P62, 55P92, 13D03, 13D07, 13D09, 13D10, 16E40, 16E45; Secondary: 57R15, 57R19, 57R56, 57R70, 57R91
Differential + Riemannian geometry
Algebraic topology
Loop space, symplectic geometry, symplectic topology, string topology, Morse theory, Hochschild and cyclic homology, operations on Hochschild and cyclic homology, rational homotopy theory, minimal models, Lagrangian embeddings, pseudo-holomorphic curves
In the late 1990s two initially unrelated developments brought free loop spaces into renewed focus. In 1999, Chas and Sullivan introduced a wealth of new algebraic operations on the homology of these spaces under the name of string topology, the full scope of which is still not completely understood. A few years earlier, Viterbo had discovered a first deep link between the symplectic topology of cotangent bundles and the topology of their free loop space. In the past 15 years, many exciting connections between these two viewpoints have been found. Still, researchers working on one side of the story often know quite little about the other. One of the main purposes of this book is to facilitate communication between topologists and symplectic geometers thinking about free loop spaces. It was written by active researchers coming to the topic from both perspectives and provides a concise overview of many of the classical results, while also beginning to explore the new directions of research that have emerged recently. As one highlight, it contains a research monograph by M. Abouzaid which proves a strengthened version of Viterbo’s isomorphism between the homology of the free loop space of a manifold and the symplectic cohomology of its cotangent bundle, following a new strategy. The book grew out of a learning seminar on free loop spaces held at Strasbourg University in 2008–2009, and should be accessible to a graduate student with a general interest in the topic. It focuses on introducing and explaining the most important aspects rather than offering encyclopedic coverage, while providing the interested reader with a broad basis for further studies and research.
9
30
2015
978-3-03719-153-8
978-3-03719-653-3
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/153
http://www.ems-ph.org/doi/10.4171/153
IRMA Lectures in Mathematics and Theoretical Physics
2523-5133
2523-5141
24
Coulomb Gases and Ginzburg–Landau Vortices
Sylvia
Serfaty
Université Pierre et Marie Curie (Paris VI), France
Statistical mechanics, structure of matter
Linear and multilinear algebra; matrix theory
Partial differential equations
82B05, 82B21, 82B26, 15B52, 82D55; 35A15, 35J20, 35J60
Statistical physics
Coulomb gas, Log gas, one-component plasma, statistical mechanics, Ginzburg-Landau, superconductivity, vortices, Abrikosov lattice, crystallization, random matrices, renormalized energy, mean-field limit, large deviations
The topic of this book is systems of points in Coulomb interaction, in particular, the classical Coulomb gas, and vortices in the Ginzburg–Landau model of superconductivity. The classical Coulomb and Log gases are classical statistical mechanics models, which have seen important developments in the mathematical literature due to their connection with random matrices and approximation theory. At low temperature, these systems are expected to “cristallize” to so-called Fekete sets, which exhibit microscopically a lattice structure. The Ginzburg–Landau model, on the other hand, describes superconductors. In superconducting materials subjected to an external magnetic field, densely packed point vortices emerge, forming perfect triangular lattice patterns, so-called Abrikosov lattices. This book describes these two systems and explores the similarity between them. It presents the mathematical tools developed to analyze the interaction between the Coulomb particles or the vortices, at the microscopic scale, and describes a “renormalized energy” governing the point patterns. This is believed to measure the disorder of a point configuration, and to be minimized by the Abrikosov lattice in dimension 2. The book gives a self-contained presentation of results on the mean field limit of the Coulomb gas system, with or without temperature, and of the derivation of the renormalized energy. It also provides a streamlined presentation of the similar analysis that can be performed for the Ginzburg–Landau model, including a review of the vortex-specific tools and the derivation of the critical fields, the mean-field limit and the renormalized energy.
3
20
2015
978-3-03719-152-1
978-3-03719-652-6
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/152
http://www.ems-ph.org/doi/10.4171/152
Zurich Lectures in Advanced Mathematics
Spectral Theory in Riemannian Geometry
Olivier
Lablée
Université Joseph Fourier Grenoble 1, Saint Martin d’Hères, France
Global analysis, analysis on manifolds
Partial differential equations
Operator theory
Calculus of variations and optimal control; optimization
58J35, 58J37, 58J50, 58J53, 35P05, 35P15, 35P20, 47A05, 47A10, 47A12, 47A60, 47A75, 49R05, 53C21
Calculus + mathematical analysis
Spectral theory, linear operators, spectrum of operators, spectral geometry, eigenvalues, Laplacian, inverse problems, Riemannian geometry, analysis on manifolds
Spectral theory is a diverse area of mathematics that derives its motivations, goals and impetus from several sources. In particular, the spectral theory of the Laplacian on a compact Riemannian manifold is a central object in differential geometry. From a physical point a view, the Laplacian on a compact Riemannian manifold is a fundamental linear operator which describes numerous propagation phenomena: heat propagation, wave propagation, quantum dynamics, etc. Moreover, the spectrum of the Laplacian contains vast information about the geometry of the manifold. This book gives a self-containded introduction to spectral geometry on compact Riemannian manifolds. Starting with an overview of spectral theory on Hilbert spaces, the book proceeds to a description of the basic notions in Riemannian geometry. Then its makes its way to topics of main interests in spectral geometry. The topics presented include direct and inverse problems. Direct problems are concerned with computing or finding properties on the eigenvalues while the main issue in inverse problems is “knowing the spectrum of the Laplacian, can we determine the geometry of the manifold?” Addressed to students or young researchers, the present book is a first introduction in spectral theory applied to geometry. For readers interested in pursuing the subject further, this book will provide a basis for understanding principles, concepts and developments of spectral geometry.
2
18
2015
978-3-03719-151-4
978-3-03719-651-9
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/151
http://www.ems-ph.org/doi/10.4171/151
EMS Textbooks in Mathematics
Hybrid Function Spaces, Heat and Navier-Stokes Equations
Hans
Triebel
University of Jena, Germany
Functional analysis
Partial differential equations
Fourier analysis
Integral equations
46-02, 46E35, 42B35, 42C40, 35K05, 35Q30, 76D03, 76D05
Differential equations
Function spaces, Morrey spaces, heat equations, Navier-Stokes equations
This book is the continuation of Local Function Spaces, Heat and Navier–Stokes Equations (Tracts in Mathematics 20, 2013) by the author. A new approach is presented to exhibit relations between Sobolev spaces, Besov spaces, and Hölder–Zygmund spaces on the one hand and Morrey–Campanato spaces on the other. Morrey–Campanato spaces extend the notion of functions of bounded mean oscillation. These spaces play a crucial role in the theory of linear and nonlinear PDEs. Chapter 1 (Introduction) describes the main motivations and intentions of this book. Chapter 2 is a selfcontained introduction into Morrey spaces. Chapter 3 deals with hybrid smoothness spaces (which are between local and global spaces) in Euclidean n-space based on the Morrey–Campanato refinement of the Lebesgue spaces. The presented approach relies on wavelet decompositions. This is applied in Chapter 4 to linear and nonlinear heat equations in global and hybrid spaces. The obtained assertions about function spaces and nonlinear heat equations are used in the Chapters 5 and 6 to study Navier–Stokes equations in hybrid and global spaces. This book is addressed to graduate students and mathematicians having a working knowledge of basic elements of (global) function spaces, and who are interested in applications to nonlinear PDEs with heat and Navier–Stokes equations as prototypes.
1
15
2015
978-3-03719-150-7
978-3-03719-650-2
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/150
http://www.ems-ph.org/doi/10.4171/150
EMS Tracts in Mathematics
24
Valuation Theory in Interaction
Antonio
Campillo
Universidad de Valladolid, Spain
Franz-Viktor
Kuhlmann
University of Saskatchewan, Saskatoon, Canada
Bernard
Teissier
Institut de Mathématiques de Jussieu, Paris, France
Field theory and polynomials
Order, lattices, ordered algebraic structures
Commutative rings and algebras
Algebraic geometry
Primary: 03CXX, 12JXX, 12E30, 12F10, 13A18, 14H20, 14M25; secondary: 06FXX, 11SXX, 11U09, 12DXX, 12E05, 12F05, 12GXX, 12L12, 13D40, 13F30, 13H05, 13JXX, 13N15, 14BXX, 14C20, 14EXX, 14F10, 14J17, 14HXX, 14PXX, 16W60, 32P05, 32SXX, 37A05, 54F50
Fields + rings
Valuation, defect, Abhyankar valuation, divisorial valuation, completion, local uniformization, toric geometry, key polynomial, excellent ring, local ring, valuation centered at a local domain, valuative tree, dicritical divisor, Rees valuation, Izumi’s theorem, plane curve singularity, Newton tree, rational surface singularity, Whitney stratification, jet scheme, embedded Nash problem, higher local field, wild ramification, dynamical system, irreducible polynomial, additive polynomial, Hilbertian field, large field, Galois theory, C-minimality, cell decomposition, imaginary element, formally real field, R-place, Hardy field, exponential-logarithmic series field, asymptotic integration, Hahn field, truncation, quasi-valuation.
Having its classical roots, since more than a century, in algebraic number theory, algebraic geometry and the theory of ordered fields and groups, valuation theory has seen an amazing expansion into many other areas in recent decades. Moreover, having been dormant for a while in algebraic geometry, it has now been reintroduced as a tool to attack the open problem of resolution of singularities in positive characteristic and to analyse the structure of singularities. Driven by this topic, and by its many new applications in other areas, also the research in valuation theory itself has been intensified, with a particular emphasis on the deep open problems in positive characteristic. As important examples for the expansion of valuation theory, it has become extremely useful in the theory of complex dynamical systems, and in the study of non-oscillating trajectories of real analytic vector fields in three dimensions. Analogues of the Riemann-Zariski valuation spaces have been found to be the proper framework for questions of intersection theory in algebraic geometry and in the analysis of singularities of complex plurisubharmonic functions. In a different direction, the relation between Berkovich geometry, tropical geometry and valuation spaces, on the one hand, and the geometry of arc spaces and valuation spaces, on the other, have begun to deepen and clarify. Ever since its beginnings, valuation theory and Galois theory have grown closely together and influenced each other. Arguably, studying and understanding the extensions of valuations in algebraic field extensions is one of the most important questions in valuation theory, whereas using valuation theory is one of he most important tools in studying Galois extensions of fields, as well as constructing field extensions with given properties. The well established topic of the model theory of valued fields is also being transformed, in particular through the study of valued fields with functions and operators, and through the study of types over valued fields. The multifaceted development of valuation theory has been monitored by two International Conferences and Workshops: the first in 1999 in Saskatoon, Canada, and the second in 2011 in Segovia and El Escorial in Spain. This book grew out of the second conference and presents high quality papers on recent research together with survey papers that illustrate the state of the art in several areas and applications of valuation theory. The book is addressed to researchers and graduate students who work in valuation theory or the areas where it is applied, as well as a general mathematical audience interested in the expansion and usefulness of the valuation theoretical approach, which has been called the “most analytic” form of algebraic reasoning. For young mathematicians who want to enter these areas of research, it provides a valuable source of up-to-date information.
9
1
2014
978-3-03719-149-1
978-3-03719-649-6
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/149
http://www.ems-ph.org/doi/10.4171/149
EMS Series of Congress Reports
2523-515X
2523-5168
A study of irreducible polynomials over henselian valued fields via distinguished pairs
Kamal
Aghigh
K.N. Toosi University of Technology, Tehran, Iran
Anuj
Bishnoi
Panjab University, Chandigarh, India
Sudesh
Khanduja
IISER, Sas Nagar, Punjab, India
Sanjeev
Kumar
Panjab University, Chandigarh, India
Valued fields, non-Archimedean valued fields, irreducible polynomials
Field theory and polynomials
General
In this paper, we give an introduction of the phenomenon of lifting with respect to residually transcendental extensions, the notion of distinguished pairs and complete distinguished chains which lead to the study of certain invariants associated to irreducible polynomials over valued fields. We give an overview of various results regarding these concepts and their applications.
1
10
1
10.4171/149-1/1
http://www.ems-ph.org/doi/10.4171/149-1/1
On fields of totally $\mathfrak{S}$-adic numbers. With an appendix by Florian Pop
Lior
Bary-Soroker
Tel Aviv University, Israel
Arno
Fehm
University of Konstanz, Germany
Totally $\mathfrak{S}$-adic numbers, Hilbertian fields
Field theory and polynomials
General
Given a finite set $\mathfrak{S}$ of places of a number field, we prove that the field of totally $\mathfrak{S}$-adic algebraic numbers is not Hilbertian.
11
15
1
10.4171/149-1/2
http://www.ems-ph.org/doi/10.4171/149-1/2
Infinite towers of Artin-Schreier defect extensions of rational function fields
Anna
Blaszczok
University of Silesia, Katowice, Poland
Defect extensions, valued rational function fields, dependent, independent Artin-Schreier defect extensions
Field theory and polynomials
Associative rings and algebras
General
We consider Artin-Scheier defect extensions of rational function fields in two variables over fields of positive characteristic. We study the problem of constructing infinite towers of such extensions. We classify Artin-Schreier defect extensions into "dependent" and "independent" ones, according to whether they are connected with purely inseparable defect extensions, or not. To understand the meaning of the classification for the issue of local uniformization, we consider various valuations of the rational function field and investigate for which it admits an infinite tower of dependent or independent Artin-Schreier defect extensions. We give also a criterion for a valued field of positive characteristic $p$ with $p$-divisible value group and perfect residue field to admit infinitely many parallel dependent Artin-Schreier defect extensions or an infinite tower of such extensions.
16
54
1
10.4171/149-1/3
http://www.ems-ph.org/doi/10.4171/149-1/3
A refinement of Izumi's Theorem
Sébastien
Boucksom
Université Paris 6, France
Charles
Favre
École Polytechnique, Palaiseau, France
Mattias
Jonsson
University of Michigan, Ann Arbor, USA
Izumi's theorem, divisorial valuation, quasimonomial valuations, volume of a valuation, toroidal embeddings, dual complexes
Commutative rings and algebras
Associative rings and algebras
Several complex variables and analytic spaces
General
We improve Izumi's inequality, which states that any divisorial valuation $v$ centered at a closed point $0$ on a normal algebraic variety $Y$ is controlled by the order of vanishing at $0$. More precisely, as $v$ ranges through valuations that are monomial with respect to coordinates in a fixed birational model $X$ dominating $Y$, we show that for any regular function $f$ on $Y$ at $0$, the function $v\mapsto v(f) / {\rm ord}_0(f)$ $d_0$ is uniformly Lipschitz continuous as a function of the weight defining $v$. As a consequence, the volume of $v$ is also a Lipschitz continuous function. Our proof uses toroidal techniques as well as positivity properties of the images of suitable nef divisors under birational morphisms.
55
81
1
10.4171/149-1/4
http://www.ems-ph.org/doi/10.4171/149-1/4
Multivariable Hodge theoretical invariants of germs of plane curves. II
Pierrette
Cassou-Noguès
Université Bordeaux I, Talence, France
Anatoly
Libgober
University of Illinois at Chicago, USA
Plane curve singularities, multivariable Alexander polynomial, faces of quasi-adjunction, spectrum of singularity, Newton trees, log-canonical thresholds
Algebraic geometry
Several complex variables and analytic spaces
General
The paper describes several invariants of plane curve singularities in terms of the data of associated Newton trees. Newton trees of singularities are discussed in detail also. The invariants which we study include the constants and faces of quasi-adjunction, log-canonical walls and Arnold-Steenbrink spectrum. As one of the consequences of these calculations we show the failure of ACC for the set of constants of quasi-adjunction of all plane curve singularities, which contains the set of log-canonical thresholds as a subset.
82
135
1
10.4171/149-1/5
http://www.ems-ph.org/doi/10.4171/149-1/5
Existence des diviseurs dicritiques, d’après S.S. Abhyankar
Vincent
Cossart
Université de Versailles Saint-Quentin, Versailles, France
Mickaël
Matusinski
Université Bordeaux 1, Talence, France
Guillermo
Moreno-Socías
CNRS/UVSQ, Versailles, France
Dicritical divisors, Rees valuations, horizontal divisors, pencil of curves
Algebraic geometry
General
In this article, there are new proofs of the existence and unicity of dicritical divisors of a pencil of plane curves of $\langle F,G\rangle$ Incidentally, we prove the equivalence between dicritical divisors and Rees valuations. Furthermore, in the case where $G_{\mathrm{red}}$ is regular at the base points of $\langle F,G\rangle$, we have that $F/G$ is residually a polynomial along any dicritical divisor; this reproves geometrically [2, Theorem (7.1)]. As a corollary of the latter proof, we get a generalization of the connectedness theorem of [8].
136
147
1
10.4171/149-1/6
http://www.ems-ph.org/doi/10.4171/149-1/6
Invariants of the graded algebras associated to divisorial valuations dominating a rational surface singularity
Vincent
Cossart
Université de Versailles Saint-Quentin, Versailles, France
Olivier
Piltant
Université de Versailles Saint-Quentin, Versailles, France
Ana
Reguera
Universidad de Valladolid, Spain
Rational surface singularity, divisorial valuation, Hilbert-Samuel function
Algebraic geometry
General
Let $(R,M)$ be a rational surface singularity and $\nu_E$ be a prime divisor of the second kind for $R$. Then $gr_{\nu_E} R$ is finitely generated over $R/M$. We recover information about the dual graph of the minimal resolution $\widetilde X$ of $\text{Spec } R$ from the set of all $gr_{\nu_E} R$. In particular we characterize those graded algebras corresponding to the exceptional curves in $\widetilde X$.
148
166
1
10.4171/149-1/7
http://www.ems-ph.org/doi/10.4171/149-1/7
An introduction to $C$-minimal structures and their cell decomposition theorem
Pablo
Cubides Kovacsics
Université Paris Diderot, Paris, France
$C$-minimality, algebraically closed valued fields, cell decomposition
Mathematical logic and foundations
Commutative rings and algebras
General
Developments in valuation theory, especially the study of algebraically closed valued fields, have used the model theory of $C$-minimal structures in different places, e.g., the work of Hrushovski-Kazdhan in [5] and Haskell-Hrushovski-Macpherson in [3]. We intend with this text both to promulgate a basic comprehension of $C$-minimality for mathematicians interested in valuation theory (equipped with a basic knowledge of model theory), and to provide a slightly different presentation of the cell decomposition theorem proved by Haskell and Macpherson in [6].
167
207
1
10.4171/149-1/8
http://www.ems-ph.org/doi/10.4171/149-1/8
Valuation semigroups of Noetherian local domains
Steven Dale
Cutkosky
University of Missouri, Columbia, United States
Valuation, Noetherian local ring, semigroup, generating sequence, defect
Commutative rings and algebras
Algebraic geometry
General
In this article we consider the problem of determining the valuation semigroup of a valuation dominating a Noetherian local ring. We give some general results and examples.
208
218
1
10.4171/149-1/9
http://www.ems-ph.org/doi/10.4171/149-1/9
Additive polynomials over perfect fields
Salih
Durhan
Middle East Technical University, Mersin, Turkey
Additive polynomials, valued fields
Number theory
General
Additive polynomials in one variable over valued fields of positive characteristic are sufficiently well understood in terms of their approximation properties. We extend results in this direction to multi-variable additive polynomials over perfect valued fields.
219
225
1
10.4171/149-1/10
http://www.ems-ph.org/doi/10.4171/149-1/10
On $\mathbb{R}$-places and related topics
Danielle
Gondard-Cozette
Université Pierre et Marie Curie, Paris, France
Formally real fields, real valuations, valuation fans, $\mathbb{R}$-places, Henselian fields, model theory of fields, real algebraic varieties, abstract spaces of orderings
Commutative rings and algebras
Mathematical logic and foundations
Field theory and polynomials
Algebraic geometry
In this survey $K$ will be a formally real fi eld, which means that $-1$ is not a finite sum of squares of elements of $K,$ hence $K$ has characteristic $0$. As often in the literature, we shall write real field insteadof formally real fieldIt is well known from Artin-Schreier theory that such fields are exactly those admitting at least one total order compatible with the field structure. After some background in Real Algebra, we introduce and study the space of $\mathbb{R}$-places. Thereafter, we present other mathematical notions, such as valuation fans, orderings of higher level and the real holomorphy ring. By use of these tools we obtain an outstanding result in Real Algebraic Geometry. Finally we provide some steps towards an abstract theory of $\mathbb{R}$-places.
226
251
1
10.4171/149-1/11
http://www.ems-ph.org/doi/10.4171/149-1/11
Extending valuations to formal completions
Francisco Javier
Herrera Govantes
Universidad de Sevilla, Spain
Miguel Ángel
Olalla Acosta
Universidad de Sevilla, Spain
Mark
Spivakovsky
Université Paul Sabatier, Toulouse, France
Bernard
Teissier
UMR 7586 du CNRS, Paris, France
Extensions of valuations, formal completion, excellent ring
Field theory and polynomials
Commutative rings and algebras
Algebraic geometry
Associative rings and algebras
This paper is an extended version of the talk given by Miguel Angel Olalla Ácosta at the International Conference on Valuation Theory in El Escorial in July 2011. Its purpose is to provide an introduction to our joint paper [5] without grinding through all of its technical details. We refer the reader to [5] for details and proofs; only a few proofs are given in the present paper.
252
265
1
10.4171/149-1/12
http://www.ems-ph.org/doi/10.4171/149-1/12
Extending real valuations to skew polynomial rings
Ángel
Granja
Universidad de León, Spain
M.
Martínez
Universidad de Valladolid, Spain
C.
Rodríguez
Universidad de Léon, Spain
Valuation, parameterized tree, real rank, factorization
Commutative rings and algebras
Field theory and polynomials
General topology
General
Let $D$ be a division ring, $T$ be a variable over $D$, $\sigma $ be an endomorphism of $D$, $\delta $ be a $\sigma$-derivation on $D$ and $R=D[T; \sigma , \delta]$ the left skew polynomial ring over $D$. We show that the partially ordered set $(Val_\nu(R),\preceq)$ of $\sigma$-compatible real valuations on $R$ extending a fixed proper real valuation $\nu $ on $D$ has a natural structure of complete parameterized non-metric tree.
266
287
1
10.4171/149-1/13
http://www.ems-ph.org/doi/10.4171/149-1/13
Stratifications in valued fields
Immanuel
Halupczok
University of Leeds, United Kingdom
Whitney stratifications, Henselian valued fields, isometries, semi-algebraic sets
Field theory and polynomials
Several complex variables and analytic spaces
General
In these notes a new, strong notion of stratifications which describe singularities of sets in Henselian valued fields is given. The first part presents the definition, some examples, and the main result about their existence. The second part explains how such a stratification in a valued field induces a classical Whitney stratification in $\mathbb{R}$.
288
296
1
10.4171/149-1/14
http://www.ems-ph.org/doi/10.4171/149-1/14
Imaginaries and definable types in algebraically closed valued fields
Ehud
Hrushovski
Hebrew University, Jerusalem, Israel
Valued fields, imaginary elements
Mathematical logic and foundations
Field theory and polynomials
General
We give an exposition of material from [1], [2] and [3], regarding definable types in the model completion of the theory of valued fields, and the classification of imaginary sorts. The latter is given a new proof, based on definable types rather than invariant types, and on the notion of generic reparametrization. I also try to bring out the relation to the geometry of [3] - stably dominated definable types as the model theoretic incarnation of a Berkovich point.
297
319
1
10.4171/149-1/15
http://www.ems-ph.org/doi/10.4171/149-1/15
Defects of algebraic function fields, completion defects and defect quotients
Franz-Viktor
Kuhlmann
University of Saskatchewan, Saskatoon, Canada
Asim
Naseem
GC University, Lahore, Pakistan
Valued function field, defect, completion, Abhyankar valuation
Field theory and polynomials
Algebraic geometry
General
The defect (also called ramification deficiency) of valued field extensions is a major stumbling block in deep open problems of valuation theory in positive characteristic. For a detailed analysis, we define and investigate two finer notions of defect: the completion defect and the defect quotient. We define all three defects for finite valued field extensions as well as for certain valued function fields (those with Abhyankar valuations that are allowed to be nontrivial on the ground field). These defects of valued function fields have played an important role in genus reduction formulas that were presented by several authors. We prove the most general known form of the Finiteness and Independence Theorem for the defect of valued function fields. Further, we investigate the completion defect and the defect quotient in detail and present analogues of the results that hold for the usual defect.
320
349
1
10.4171/149-1/16
http://www.ems-ph.org/doi/10.4171/149-1/16
On generalized series fields and exponential-logarithmic series fields with derivations
Mickaël
Matusinski
Université Bordeaux 1, Talence, France
Hardy fields, generalized series fields and exponential-logarithmic series fields with derivations, asymptotic integration, integration
Field theory and polynomials
Commutative rings and algebras
General
We survey some important properties of fields of generalized series and of exponential-logarithmic series, with particular emphasis on their possible di fferential structure, based on a joint work of the author with S. Kuhlmann [40, 39].
350
372
1
10.4171/149-1/17
http://www.ems-ph.org/doi/10.4171/149-1/17
Jet schemes of rational double point singularities
Hussein
Mourtada
Institut Mathématique de Jussieu-Paris Rive Gauche, Paris, France
Jet schemes, embedded Nash problem, rational double point singularities, Hilbert-Poincaré series
Algebraic geometry
Commutative rings and algebras
General
We prove that for $m\in \mathbb{N},~m$ large enough, the number of irreducible components of the schemes of $m-$jets centered at a point which is a double point singularity is independent of $m$ and is equal to the number of exceptional curves on the minimal resolution of the singularity. We also relate some irreducible components of the jet schemes of an $E_6$ singularity to its "minimal" embedded resolutions of singularities.
373
388
1
10.4171/149-1/18
http://www.ems-ph.org/doi/10.4171/149-1/18
Valuations centered at a two-dimensional regular local domain: infima and topologies
Josnei
Novacoski
University of Silesia, Katowice, Poland
Valuative tree, non-metric tree, valuations centered at a local domain
Commutative rings and algebras
Algebraic geometry
General
Take a two-dimensional regular local domain $R$. The space of all valuations centered at $R$ has a non-metric tree structure, called the valuative tree of $R$. However, the notion of non-metric tree appearing in the literature does not guarantee the existence of infimum for a non-empty set of valuations. We give a more general definition of a rooted non-metric tree and prove that the valuative tree has this more general property. We also generalize some topological results related to a non-metric tree. For instance, we show that the weak tree topology is always coarser than the metric topology given by any parametrization.
389
403
1
10.4171/149-1/19
http://www.ems-ph.org/doi/10.4171/149-1/19
Reduction of local uniformization to the rank one case
Josnei
Novacoski
University of Silesia, Katowice, Poland
Mark
Spivakovsky
Université Paul Sabatier, Toulouse, France
Local uniformization, rank one valuations, valuations centered at a local ring
Commutative rings and algebras
Algebraic geometry
General
The main result of this paper is that in order to prove the local uniformization theorem for local domains, it is enough to prove it for rank one valuations. Our proof does not depend on the nature of the class of local domains for which we want to prove local uniformization. We prove also the reductions for diff erent versions of the local uniformization theorem.
404
431
1
10.4171/149-1/20
http://www.ems-ph.org/doi/10.4171/149-1/20
Little survey on large fields - old & new
Florian
Pop
University of Pennsylvania, Philadelphia, United States
Large fields, ultraproducts, PAC, pseudo closed fields, Henselian pairs, elementary equivalence, algebraic varieties, rational points, function fields, (inverse) Galois theory, embedding problems, model theory, rational connectedness, extremal fields
Field theory and polynomials
General
The large elds were introduced by the author in [60] and subsequently acquired several other names. This little survey includes earlier and new developments, and at the end of each section we mention a few open questions.
432
463
1
10.4171/149-1/21
http://www.ems-ph.org/doi/10.4171/149-1/21
Quasi-valuations -- topology and the weak approximation theorem
Shai
Sarussi
Sce College, Ashdod, Israel
Quasi-valuations, approximation theorem
Commutative rings and algebras
General topology
General
Suppose $F$ is a field with a nontrivial valuation $v$ and valuation ring $O_{v}$, $E$ is a finite field extension and $w$ is a quasi-valuation on $E$ extending $v$. We study the topology induced by $w$. We prove that the quasi-valuation ring determines the topology, independent of the choice of its quasi-valuation. Moreover, we prove the weak approximation theorem for quasi-valuations.
464
473
1
10.4171/149-1/22
http://www.ems-ph.org/doi/10.4171/149-1/22
Overweight deformations of affine toric varieties and local uniformization
Bernard
Teissier
UMR 7586 du CNRS, Paris, France
Toric geometry, valuations, uniformization, key polynomials
Algebraic geometry
General
Given an equicharacteristic complete noetherian local ring $R$ with algebraically closed residue field $k$, we first present a combinatorial proof of \emph{embedded} local uniformization for zero-dimensional valuations of $R$ whose associated graded ring ${\rm gr}_\nu R$ with respect to the filtration defined by the valuation is a finitely generated $k$-algebra. The main idea here is that some of the birational toric maps which provide embedded pseudo-resolutions for the affine toric variety corresponding to ${\rm gr}_\nu R$ also provide local uniformizations for $\nu$ on $R$. These valuations are necessarily Abhyankar (for zero-dimensional valuations this means that the value group is $\mathbf Z^r$ with $r={\rm dim}R$).\Par In a second part we show that conversely, given an excellent noetherian equicharacteristic local domain $R$ with algebraically closed residue field, if the zero-dimensional valuation $\nu$ of $R$ is Abhyankar, there are local domains $R'$ which are essentially of finite type over $R$ and dominated by the valuation ring $R_\nu$ ($\nu$-modifications of $R$) such that the semigroup of values of $\nu$ on $R'$ is finitely generated, and therefore so is the $k$-algebra ${\rm gr}_\nu R'$. Combining the two results and using the fact that Abhyankar valuations behave well under completion gives a proof of local uniformization for rational Abhyankar valuations and, by a specialization argument, for all Abhyankar valuations. \par As a by-product we obtain a description of the valuation ring of a rational Abhyankar valuation as an inductive limit indexed by $\mathbf N$ of birational toric maps of regular local rings. One of our main tools, the valuative Cohen theorem, is then used to study the extensions of rational monomial Abhyankar valuations of the ring $k[[x_1,\ldots ,x_r]]$ to monogenous integral extensions and the nature of their key polynomials. In the conclusion we place the results in the perspective of local embedded resolution of singularities by a single toric modification after an appropriate re-embedding.
474
565
1
10.4171/149-1/23
http://www.ems-ph.org/doi/10.4171/149-1/23
Detecting valuations using small Galois groups
Adam
Topaz
University of California, Berkeley, USA
Valuation theory, pro-$\ell$ Galois groups, abelian-by-central, local theory
Field theory and polynomials
General
In this note we show how to detect valuations using almost-abelian pro-$\ell$ Galois groups of a field. In particular, we show that "commuting-liftable" subgroups of Galois groups arise, in a controlled way, from Kummer-duals of (principal-)units of valuations.
568
578
1
10.4171/149-1/24
http://www.ems-ph.org/doi/10.4171/149-1/24
Truncation in Hahn fields
Lou
van den Dries
University of Illinois at Urbana-Champaign, USA
Hahn fields, truncation
Order, lattices, ordered algebraic structures
Field theory and polynomials
General
We consider truncation closed subgroups, subrings, and subfi elds of Hahn fields, and show that the property of being truncation closed is preserved under various natural ways of extending these substructures.
579
595
1
10.4171/149-1/25
http://www.ems-ph.org/doi/10.4171/149-1/25
The ergodicity of 1-Lipschitz transformations on 2-adic spheres
Ekaterina
Yurova
Linnaeus University, Vaxjo, Sweden
Dynamical systems, ergodicity, $P$-adic spheres, Van der Put series
Dynamical systems and ergodic theory
General
In this paper we present results about ergodicity of dynamical systems on $2$-adic spheres for 1-Lipschitz maps $f:\mathbb Z_2\rightarrow \mathbb Z_2$ announced in [8], and extension of Theorem 3 from [8] for the case of spheres of radii greater than $\frac{1}{8}.$ We propose a new approach to study ergodic properties of 1-Lipschitz transformations of $2$-adic spheres. We use a representation of continuous functions $f$ via its van der Put series. This technique allows us to go beyond the classes of smooth 1-Lipschitz transformations which were studied earlier.
596
599
1
10.4171/149-1/26
http://www.ems-ph.org/doi/10.4171/149-1/26
Ramification of higher local fields approaches and questions
Liang
Xiao
University of California at Irvine, USA
Igor
Zhukov
St. Petersburg University, Russian Federation
Complete discrete valuation field, higher local field, imperfect residue field, wild ramification, Swan conductor, Artin conductor
Algebraic geometry
Number theory
General
A survey paper contains facts, ideas and problems related to ramifi cation in fi nite extensions of complete discrete valuation fi elds with arbitrary residue fields. Some new results are included.
600
656
1
10.4171/149-1/27
http://www.ems-ph.org/doi/10.4171/149-1/27
Sophus Lie and Felix Klein: The Erlangen Program and Its Impact in Mathematics and Physics
Lizhen
Ji
University of Michigan, Ann Arbor, USA
Athanase
Papadopoulos
Université de Strasbourg, France
History and biography
Topological groups, Lie groups
Geometry
Differential geometry
01-00, 01-02, 01A05, 01A55, 01A70, 22-00, 22-02, 22-03, 51N15, 51P05, 53A20, 53A35, 53B50, 54H15, 58E40
History of mathematics
Geometry
Sophus Lie, Felix Klein, the Erlangen program, group action, Lie group action, symmetry, projective geometry, non-Euclidean geometry, spherical geometry, hyperbolic geometry, transitional geometry, discrete geometry, transformation group
The Erlangen program expresses a fundamental point of view on the use of groups and transformation groups in mathematics and physics. The present volume is the first modern comprehensive book on that program and its impact in contemporary mathematics and physics. Klein spelled out the program, and Lie, who contributed to its formulation, is the first mathematician who made it effective in his work. The theories that these two authors developed are also linked to their personal history and to their relations with each other and with other mathematicians, incuding Hermann Weyl, Élie Cartan, Henri Poincaré, and many others. All these facets of the Erlangen program appear in the present volume. The book is written by well-known experts in geometry, physics and history of mathematics and physics. It is addressed to mathematicians, to graduate students, and to all those interested in the development of mathematical ideas.
4
30
2015
978-3-03719-148-4
978-3-03719-648-9
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/148
http://www.ems-ph.org/doi/10.4171/148
IRMA Lectures in Mathematics and Theoretical Physics
2523-5133
2523-5141
23
Sophus Lie, a giant in mathematics
Lizhen
Ji
University of Michigan, Ann Arbor, USA
History and biography
General
1
26
1
10.4171/148-1/1
http://www.ems-ph.org/doi/10.4171/148-1/1
Felix Klein: his life and mathematics
Lizhen
Ji
University of Michigan, Ann Arbor, USA
History and biography
General
27
58
1
10.4171/148-1/2
http://www.ems-ph.org/doi/10.4171/148-1/2
Klein and the Erlangen Programme
Jeremy
Gray
The Open University, Milton Keynes, UK
History and biography
General
59
75
1
10.4171/148-1/3
http://www.ems-ph.org/doi/10.4171/148-1/3
Klein’s “Erlanger Programm”: do traces of it exist in physical theories?
Hubert
Goenner
Universität Göttingen, Germany
History and biography
General
77
90
1
10.4171/148-1/4
http://www.ems-ph.org/doi/10.4171/148-1/4
On Klein’s So-called Non-Euclidean geometry
Norbert
A’Campo
Universität Basel, Switzerland
Athanase
Papadopoulos
Université de Strasbourg, France
History and biography
General
91
136
1
10.4171/148-1/5
http://www.ems-ph.org/doi/10.4171/148-1/5
What are symmetries of PDEs and what are PDEs themselves?
Alexandre
Vinogradov
Lizzano in Belvedere (Bo), Italy
History and biography
General
137
190
1
10.4171/148-1/6
http://www.ems-ph.org/doi/10.4171/148-1/6
Transformation groups in non-Riemannian geometry
Charles
Frances
Université Paris-Sud, Orsay, France
History and biography
General
191
216
1
10.4171/148-1/7
http://www.ems-ph.org/doi/10.4171/148-1/7
Transitional geometry
Norbert
A’Campo
Universität Basel, Switzerland
Athanase
Papadopoulos
Université de Strasbourg, France
History and biography
General
217
235
1
10.4171/148-1/8
http://www.ems-ph.org/doi/10.4171/148-1/8
On the projective geometry of constant curvature spaces
Athanase
Papadopoulos
Université de Strasbourg, France
Sumio
Yamada
Gakushuin University, Tokyo, Japan
History and biography
General
237
245
1
10.4171/148-1/9
http://www.ems-ph.org/doi/10.4171/148-1/9
The Erlangen program and discrete differential geometry
Yuri
Suris
Technische Universität Berlin, Germany
History and biography
General
247
281
1
10.4171/148-1/10
http://www.ems-ph.org/doi/10.4171/148-1/10
Three-dimensional gravity – an application of Felix Klein’s ideas in physics
Catherine
Meusburger
Universität Erlangen-Nürnberg, Germany
History and biography
General
283
306
1
10.4171/148-1/11
http://www.ems-ph.org/doi/10.4171/148-1/11
Invariances in physics and group theory
Jean-Bernard
Zuber
Universite Pierre et Marie Curie, Paris, France
History and biography
General
307
324
1
10.4171/148-1/12
http://www.ems-ph.org/doi/10.4171/148-1/12
Handbook of Hilbert Geometry
Athanase
Papadopoulos
Université de Strasbourg, France
Marc
Troyanov
École Polytechnique Fédérale de Lausanne, Switzerland
Differential geometry
Geometry
Convex and discrete geometry
Global analysis, analysis on manifolds
01A55, 01-99, 35Q53, 37D25, 37D20, 37D40, 47H09, 51-00, 51-02, 51-03, 51A05, 51B20, 51F99, 51K05, 51K10, 51K99, 51M10, 52A07, 52A20, 52A99, 53A20, 53A35, 53B40, 53C22, 53C24, 53C60, 53C70, 53B40, 54H20, 57S25, 58-00, 58-02, 58-03, 58B20, 58D05, 58F07.
Differential + Riemannian geometry
Hilbert metric, Funk metric, non-symmetric metric, Finsler geometry, Minkowski space, Minkowski functional, convexity, Cayley-Klein-Beltrami model, projective manifold, projective volume, Busemann curvature, Busemann volume, horofunction, geodesic flow, Teichmüller space, Hilbert fourth problem, entropy, geodesic, Perron-Frobenius theory, geometric structure, holonomy homomorphism
This volume presents surveys, written by experts in the field, on various classical and the modern aspects of Hilbert geometry. They are assuming several points of view: Finsler geometry, calculus of variations, projective geometry, dynamical systems, and others. Some fruitful relations between Hilbert geometry and other subjects in mathematics are emphasized, including Teichmüller spaces, convexity theory, Perron–Frobenius theory, representation theory, partial differential equations, coarse geometry, ergodic theory, algebraic groups, Coxeter groups, geometric group theory, Lie groups and discrete group actions. The Handbook is addressed to both students who want to learn the theory and researchers working in the area.
12
1
2014
978-3-03719-147-7
978-3-03719-647-2
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/147
http://www.ems-ph.org/doi/10.4171/147
IRMA Lectures in Mathematics and Theoretical Physics
2523-5133
2523-5141
22
Weak Minkowski spaces
Athanase
Papadopoulos
Université de Strasbourg, France
Marc
Troyanov
Ecole Polytechnique Fédérale de Lausanne, Switzerland
Weak Minkowski space, Minkowski geometry, norm, Hilbert geometry, Funk geometry, weak norm, Mazur–Ulam theorem, Desarguesian space, Busemann G-space
Geometry
Differential geometry
General
We define the notion of weak Minkowski metric and prove some basic properties of such metrics. We also highlight some of the important analogies between Minkowski geometry and the Funk and Hilbert geometries.
11
32
1
10.4171/147-1/1
http://www.ems-ph.org/doi/10.4171/147-1/1
From Funk to Hilbert geometry
Athanase
Papadopoulos
Université de Strasbourg, France
Marc
Troyanov
Ecole Polytechnique Fédérale de Lausanne, Switzerland
Funk metric, convexity, Hilbert metric, Busemann's methods
Geometry
Differential geometry
General
We survey some basic geometric properties of the Funk metric of a convex set in $\mathbb{R}^n$. In particular, we study its geodesics, its topology, its metric balls, its convexity properties, its perpendicularity theory and its isometries. The Hilbert metric is a symmetrization of the Funk metric, and we show some properties of the Hilbert metric that follow directly from the properties we prove for the Funk metric.
33
67
1
10.4171/147-1/2
http://www.ems-ph.org/doi/10.4171/147-1/2
Funk and Hilbert geometries from the Finslerian viewpoint
Marc
Troyanov
Ecole Polytechnique Fédérale de Lausanne, Switzerland
Hilbert geometry, Funk geometry, Finlser metric, flag curvature, projective metric
Differential geometry
General
In 1929, Paul Funk and Ludwig Berwald gave a characterization of Hilbert geometries from the Finslerian viewpoint. They showed that a smooth Finsler metric in a strongly convex bounded domain of $\mathbb{R}^n$ is the Hilbert geometry in that domain if and only if it is complete, if its geodesics are straight lines and if its flag curvature is equal to $-1$. The goal of this chapter is to explain these notions in details, to illustrate the relation between Hilbert geometry, Finsler geometry and the calculus of variations, and to prove the Funk–Berwald characterization theorem.
69
110
1
10.4171/147-1/3
http://www.ems-ph.org/doi/10.4171/147-1/3
On the Hilbert geometry of convex polytopes
Constantin
Vernicos
Université Montpellier 2, France
Hilbert geometry, Finsler geometry, metric spaces, normed vector spaces, Lipschitz distance
Differential geometry
Geometry
General
We survey the Hilbert geometry of convex polytopes. In particular we present two important characterizations of these geometries, the first one in terms of the volume growth of their metric balls, the second one as a metric bilipschitzly equivalent to the Hilbert geometry in the simplex.
111
125
1
10.4171/147-1/4
http://www.ems-ph.org/doi/10.4171/147-1/4
The horofunction boundary and isometry group of the Hilbert geometry
Cormac
Walsh
Ecole Polytechnique, Palaiseau, France
Horofunction boundary, Busemann function, detour cost, Hilbert isometry
Convex and discrete geometry
General
The horofunction boundary is a means of compactifying metric spaces that was introduced by Gromov in the 1970s. We describe explicitly the horofunction boundary of the Hilbert geometry, and sketch how it may be used to study the isometry group of this space.
127
146
1
10.4171/147-1/5
http://www.ems-ph.org/doi/10.4171/147-1/5
Characterizations of hyperbolic geometry among Hilbert geometries
Ren
Guo
Oregon State University, Corvallis, USA
Hilbert geometry, hyperbolic geometry, Finsler structure
Geometry
Convex and discrete geometry
Differential geometry
General
This chapter is a survey of different characterizations of hyperbolic geometry among Hilbert geometries.
147
158
1
10.4171/147-1/6
http://www.ems-ph.org/doi/10.4171/147-1/6
Around groups in Hilbert geometry
Ludovic
Marquis
Université de Rennes I, France
Flow completion, Burgers equation, manifolds of mappings
Global analysis, analysis on manifolds
Partial differential equations
General
In this chapter, we survey groups of projective transformations acting on a Hilbert geometry.
207
261
1
10.4171/147-1/7
http://www.ems-ph.org/doi/10.4171/147-1/7
The geodesic flow of Finsler and Hilbert geometries
Mickaël
Crampon
Universidad de Santiago de Chile, Chile
Geodesic flows, Finsler geometry, Hilbert geometry, hyperbolic dynamics, Lyapunov exponents, entropy
Dynamical systems and ergodic theory
Differential geometry
General
This is a survey of the dynamics of the geodesic flow of Hilbert geometries. The main idea is to compare this flow with the geodesic flow of negatively curved Finsler or Riemannian manifolds, by making links between various useful objects and by comparing results and questions.
161
206
1
10.4171/147-1/8
http://www.ems-ph.org/doi/10.4171/147-1/8
Dynamics of Hilbert nonexpansive maps
Anders
Karlsson
Université de Genève, Switzerland
Hilbert metric, non-expansive maps
Operator theory
Geometry
General
In his work on the foundations of geometry, David Hilbert observed that a formula which appeared in works by Klein gives rise to a complete metric on any bounded convex domain. Some decades later, Garrett Birkhoff and Hans Samelson noticed that this metric has interesting applications, when considering certain maps of convex cones that contract the metric. Such situations have since arisen in many contexts, pure and applied, and could be called nonlinear Perron–Frobenius theory. This note centers around one dynamical aspect of this theory.
263
273
1
10.4171/147-1/9
http://www.ems-ph.org/doi/10.4171/147-1/9
Birkhoff’s version of Hilbert’s metric and its applications in analysis
Bas
Lemmens
University of Kent, Canterbury, United Kingdom
Roger
Nussbaum
Rutgers University, Piscataway, USA
Birkhoff's version of Hilbert's metric, Birkhoff's contraction coefficient, Denjoy–Wolff type theorems, dynamics of non-expansive mappings, isometric embeddings, nonlinear mappings on cones
Operator theory
Differential geometry
General topology
General
Birkhoff's version of Hilbert's metric is a distance between pairs of rays in a closed cone, and is closely related to Hilbert's classical cross-ratio metric. The version we discuss here was popularized by Bushell and can be traced back to the work of Garrett Birkhoff and Hans Samelson. It has found numerous applications in mathematical analysis, especially in the analysis of linear, and nonlinear, mappings on cones. Some of these applications are discussed in this chapter. Birkhoff's version of Hilbert's metric provides a different perspective on Hilbert geometries and naturally leads to infinite-dimensional generalizations. We illustrate this by showing some of its uses in the geometric analysis of Hilbert geometries.
275
303
1
10.4171/147-1/10
http://www.ems-ph.org/doi/10.4171/147-1/10
Convex real projective structures and Hilbert metrics
Inkang
Kim
KIAS, Seoul, South Korea
Athanase
Papadopoulos
Université de Strasbourg, France
Convex real projective structure, geodesic flow, deformation space, hyperbolic structure, geodesic current, topological entropy, volume entropy, Busemann cocycle, Patterson–Sullivan measure
Geometry
Manifolds and cell complexes
General
We review some basic concepts related to convex real projective structures from the differential geometry point of view. We start by recalling a Riemannian metric which originates in the study of affine spheres using the Blaschke connection (work of Calabi and of Cheng--Yau) mentioning its relation with the Hilbert metric. We then survey some of the deformation theory of convex real projective structures on surfaces. We describe in particular how the set of (Hilbert) lengths of simple closed curves is used in a parametrization of the deformation space in analogy with the classical Fenchel–Nielsen parameters of Teichmüller space (work of Goldman). We then mention parameters of this deformation space that arise in the work of Hitchin on the character variety of representations of the fundamental group of the surface in $\mathrm{SL}(3,\mathbb{R})$. In this character variety, the component of the character variety that corresponds to projective structures is identified with the vector space of pairs of holomorphic quadratic and cubic differentials over a fixed Riemann surface. Labourie and Loftin (independently) obtained parameter spaces that use the cubic differentials and affine spheres. We then display some similarities and differences between Hilbert geometry and hyperbolic geometry using geodesic currents and topological entropy. Finally, we discuss geodesic flows associated to Hilbert metrics and compactifications of spaces of convex real projective structures on surfaces. This makes another analogy with works done on the Teichmüller space of the surface.
307
338
1
10.4171/147-1/11
http://www.ems-ph.org/doi/10.4171/147-1/11
Weil–Petersson Funk metric on Teichmüller space
Hideki
Miyachi
Osaka University, Japan
Ken’ichi
Ohshika
Osaka University Graduate School of Science, Japan
Sumio
Yamada
Gakushuin University, Tokyo, Japan
Teichmüller space, Weil–Petersson metric, Funk metric, convex geometry
Global analysis, analysis on manifolds
Partial differential equations
General
As a deformation space of hyperbolic metrics defined on a closed surface of genus $g \geq 2$, Teichmüller space can be regarded as a convex set with respect to the Weil--Petersson geometry. The convexity is used to construct a new distance function, called the Weil–Petersson Funk metric, which is a weak metric, lacking the symmetry and non-degeneracy conditions. The distance between two points is defined as a supremum of functions on the pair of points indexed by the set of the simple closed curves of the given surface. This set acts as the index set of the supporting hyperplanes of Teichmüller space regarded as a Weil–Petersson convex body. This Funk-type construction also appears in defining the two well-known distance functions on Teichmüller space: the Teichmüller metric and the Thurston metric. We emphasize in this chapter that the underlying idea for the three distance functions is to treat the Teichmüller space as a convex body.
339
352
1
10.4171/147-1/12
http://www.ems-ph.org/doi/10.4171/147-1/12
Funk and Hilbert geometries in spaces of constant curvature
Athanase
Papadopoulos
Université de Strasbourg, France
Sumio
Yamada
Gakushuin University, Tokyo, Japan
Hilbert metric, Funk metric, constant curvature, trigonometry
Global analysis, analysis on manifolds
Geometry
Differential geometry
General
We survey the Funk and Hilbert geometries of open convex sets in the sphere $S^n$ and in the hyperbolic space $\mathbb{H}^n$. The theories are developed in analogy with the classical theory in Euclidean space. It is rather unexpected that the Funk geometry, whose definition and development use the affine structure of Euclidean space, has analogues in the non-linear spaces $S^n$ and $\mathbb{H}^n$, where there are no analogies. As we shall see, the existence of a Funk geometry in these non-linear spaces is based on some non-Euclidean trigonometric formulae which display some kind of similarity between (the hyperbolic sine of the lengths of) sides of right triangles. The Hilbert metric in each of the constant curvature settings is a symmetrization of the Funk metric. We show that the Hilbert metric of a convex subset in a space of constant curvature can also be defined using a notion of a cross ratio which is proper to that space.
353
379
1
10.4171/147-1/13
http://www.ems-ph.org/doi/10.4171/147-1/13
On the origin of Hilbert geometry
Marc
Troyanov
Ecole Polytechnique Fédérale de Lausanne, Switzerland
David Hilbert , Hilbert geometry
Geometry
History and biography
General
In this brief chapter we succinctly comment on the historical origin of Hilbert geometry. In particular, we give a summary of the letter in which David Hilbert informs his friend and colleague Felix Klein about his discovery of this geometry.
383
389
1
10.4171/147-1/14
http://www.ems-ph.org/doi/10.4171/147-1/14
Hilbert’s fourth problem
Athanase
Papadopoulos
Université de Strasbourg, France
Hilbert problems, Busemann geometry, Hilbert's Problem IV, Crofton formula, Hilbert metric, Desarguesian space, projective metric
History and biography
Geometry
Differential geometry
Global analysis, analysis on manifolds
Hilbert's fourth problem asks for the construction and the study of metrics on subsets of projective space for which the projective line segments are geodesics. Several solutions of the problem were given so far, depending on more precise interpretations of this problem, with various additional conditions satisfied. The most interesting solutions are probably those inspired by an integral formula that was first introduced in this theory by Herbert Busemann. Besides that, Busemann and his school made a thorough investigation of metrics defined on subsets of projective space for which the projective lines are geodesics and they obtained several results, characterizing several classes of such metrics. We review some of the developments and important results related to Hilbert's problem, especially those that arose from Busemann's work, mentioning recent results and connections with several branches of mathematics, including Riemannian geometry, the foundations of mathematics, the calculus of variations, metric geometry and Finsler geometry. Hilbert metrics – the subject of this handbook – constitute a basic class of metrics that satisfy the requirements of Hilbert's problem.
391
431
1
10.4171/147-1/15
http://www.ems-ph.org/doi/10.4171/147-1/15
Open problems
General
433
442
1
10.4171/147-1/16
http://www.ems-ph.org/doi/10.4171/147-1/16
Emil Artin and Beyond – Class Field Theory and $L$-Functions
Della
Dumbaugh
University of Richmond, USA
Joachim
Schwermer
University of Vienna, Austria
History and biography
Number theory
01A60, 01A70, 11R37, 11R39, 11S37, 11S39, 11Fxx, 11Mxx
History of mathematics
Number theory, class field theory, L-functions, automorphic L-functions; history of mathematics, Emil Artin
This book explores the development of number theory, and class field theory in particular, as it passed through the hands of Emil Artin, Claude Chevalley and Robert Langlands in the middle of the twentieth century. Claude Chevalley’s presence in Artin’s 1931 Hamburg lectures on class field theory serves as the starting point for this volume. From there, it is traced how class field theory advanced in the 1930s and how Artin’s contributions influenced other mathematicians at the time and in subsequent years. Given the difficult political climate and his forced emigration as it were, the question of how Artin created a life in America within the existing institutional framework, and especially of how he continued his education of and close connection with graduate students, is considered. In particular, Artin’s collaboration in algebraic number theory with George Whaples and his student Margaret Matchett’s thesis work “On the zeta-function for ideles” in the 1940s are investigated. A (first) study of the influence of Artin on present day work on a non-abelian class field theory finishes the book. The volume consists of individual essays by the authors and two contributors, James Cogdell and Robert Langlands, and contains relevant archival material. Among these, the letter from Chevalley to Helmut Hasse in 1935 is included, where he introduces the notion of ideles and explores their significance, along with the previously unpublished thesis by Matchett and the seminal letter of Langlands to André Weil of 1967 where he lays out his ideas regarding a non-abelian class field theory. Taken together, these chapters offer a view of both the life of Artin in the 1930s and 1940s and the development of class field theory at that time. They also provide insight into the transmission of mathematical ideas, the careful steps required to preserve a life in mathematics at a difficult moment in history, and the interplay between mathematics and politics (in more ways than one). Some of the technical points in this volume require a sophisticated understanding of algebra and number theory. The broader topics, however, will appeal to a wider audience that extends beyond mathematicians and historians of mathematics to include historically minded individuals, particularly those with an interest in the time period.
3
31
2015
978-3-03719-146-0
978-3-03719-646-5
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/146
http://www.ems-ph.org/doi/10.4171/146
Heritage of European Mathematics
2523-5214
2523-5222
Uniformization of Riemann Surfaces
Revisiting a hundred-year-old theoremTranslated from the French by Robert G. Burns
Henri Paul
de Saint-Gervais
Paris
History and biography
General
(primary; secondary): 30F10; 01A55, 14H55, 30-01, 30-03, 30Fxx
History of mathematics
Riemann surfaces, uniformization, Fuchsian groups, continuity method, Gauss, Riemann, Schwarz, Poincaré, Klein, Koebe
In 1907 Paul Koebe and Henri Poincar é almost simultaneously proved the uniformization theorem: Every simply connected Riemann surface is isomorphic to the plane, the open unit disc, or the sphere. It took a whole century to get to the point of stating this theorem and providing a convincing proof of it, relying as it did on prior work of Gauss, Riemann, Schwarz, Klein, Poincar é, and Koebe, among others. The present book o ffers an overview of the maturation process of this theorem. The evolution of the uniformization theorem took place in parallel with the emergence of modern algebraic geometry, the creation of complex analysis, the fi rst stirrings of functional analysis, and with the flowering of the theory of di fferential equations and the birth of topology. The uniformization theorem was thus one of the lightning rods of 19th century mathematics. Rather than describe the history of a single theorem, our aim is to return to the original proofs, to look at these through the eyes of modern mathematicians, to enquire as to their correctness, and to attempt to make them rigorous while respecting insofar as possible the state of mathematical knowledge at the time, or, if this should prove impossible, then using modern mathematical tools not available to their authors. This book will be useful to today's mathematicians wishing to cast a glance back at the history of their discipline. It should also provide graduate students with a non-standard approach to concepts of great importance for modern research.
1
31
2016
978-3-03719-145-3
978-3-03719-645-8
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/145
http://www.ems-ph.org/doi/10.4171/145
Heritage of European Mathematics
2523-5214
2523-5222
Karl Löwner and His Student Lipman Bers – Pre-war Prague Mathematicians
Martina
Bečvářová
Czech Technical University, Prague, Czech Republic
Ivan
Netuka
Charles University, Prague, Czech Republic
History and biography
Number theory
01A60, 01A70, 11R37, 11R39, 11S37, 11S39, 11Fxx, 11Mxx
History of mathematics
Number theory
Mathematical analysis, matrix functions, geometric function theory, potential theory, 20th century history of mathematics
This monograph is devoted to two distinguished mathematicians, Karel Löwner (1893–1968) and Lipman Bers (1914–1993), whose lives are dramatically interlinked with key historical events of the 20th century. K. Löwner, Professor of Mathematics at the German University in Prague (Czechoslovakia), was dismissed from his position because he was a Jew, and emigrated to the USA in 1939 (where he changed his name to Charles Loewner). Earlier, he had published several outstanding papers in complex analysis and a masterpiece on matrix functions. In particular, his ground-breaking parametric method in geometric function theory from 1923, which led to Löwner’s celebrated differential equation, brought him world-wide fame and turned out to be a cornerstone in de Branges’ proof of the Bieberbach conjecture. Unexpectedly, Löwner’s differential equation has gained recent prominence with the introduction of a conformally invariant stochastic process called stochastic Loewner evolution (SLE) by O. Schramm in 2000. SLE features in two Fields Medal citations from 2006 and 2010. L. Bers was the final Prague Ph.D. student of K. Löwner. His dissertation on potential theory (1938), completed shortly before his emigration and long thought to be irretrievably lost, was found in 2006. It is here made accessible for the first time, with an extensive commentary, to the mathematical community. This monograph presents an in-depth account of the lives of both mathematicians, with special emphasis on the pre-war period. Löwner’s teaching activities and professional achievements are presented in the context of the prevailing complex political situation and against the background of the wider development of mathematics in Europe. Each of his publications is accompanied by an extensive commentary, tracing the origin and motivation of the problem studied, and describing the state-of-art at the time of the corresponding mathematical field. Special attention is paid to the impact of the results obtained and to the later development of the underlying ideas, thus connecting Löwner’s achievements to current research activity. The text is based on an extensive archival search, and most of the archival findings appear here for the first time. Anyone with an interest in mathematics and the history of mathematics will enjoy reading this book about two famous mathematicians of the 20th century.
4
10
2015
978-3-03719-144-6
978-3-03719-644-1
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/144
http://www.ems-ph.org/doi/10.4171/144
Heritage of European Mathematics
2523-5214
2523-5222
Faà di Bruno Hopf Algebras, Dyson–Schwinger Equations, and Lie–Butcher Series
Kurusch
Ebrahimi-Fard
Universidad Autónoma de Madrid, Spain
Frédéric
Fauvet
Université de Strasbourg, France
Combinatorics
Order, lattices, ordered algebraic structures
Associative rings and algebras
Approximations and expansions
Primary: 05E15, 06A07, 16T05, 41A58, 58D05, 93C10; Secondary: 05C05, 81T18, 34A25, 34M25, 47H20, 65L05, 81T15, 81T16
Combinatorics + graph theory
Lattice theory
Fields + rings
Linear algebra
Faà di Bruno formula, Dyson–Schwinger equations, geometric numerical integration, Butcher series, Lie–Butcher series, nonlinear control systems, nonlinear operators, combinatorial Hopf algebras, pre-Lie algebras, Lie algebras, trees, Ecalle’s mould calcul
Since the early works of G.-C. Rota and his school, Hopf algebras have been instrumental in algebraic combinatorics. In a seminal 1998 paper, A. Connes and D. Kreimer presented a Hopf algebraic approach to renormalization in perturbative Quantum Field Theory (QFT). This work triggered an abundance of new research on applications of Hopf algebraic techniques in QFT as well as other areas of theoretical physics. Furthermore, these new developments were complemented by progress made in other domains of applications, such as control theory, dynamical systems, and numerical integration methods. Especially in the latter context, it became clear that J. Butcher’s work from the early 1970s was well ahead of its time. The present volume emanated from a conference hosted in June 2011 by IRMA at Strasbourg University in France. Researchers from different scientific communities who share similar techniques and objectives gathered at this meeting to discuss new ideas and results on Faà di Bruno algebras, Dyson–Schwinger equations, and Butcher series. The purpose of this book is to present a coherent set of lectures reflecting the state of the art of research on combinatorial Hopf algebras relevant to high energy physics, control theory, dynamical systems, and numerical integration methods. More specifically, connections between Dyson–Schwinger equations, Faà di Bruno algebras, and Butcher series are examined in great detail. This volume is aimed at researchers and graduate students interested in combinatorial and algebraic aspects of QFT, control theory, dynamical systems and numerical analysis of integration methods. It contains introductory lectures on the various constructions that are emerging and developing in these domains.
6
30
2015
978-3-03719-143-9
978-3-03719-643-4
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/143
http://www.ems-ph.org/doi/10.4171/143
IRMA Lectures in Mathematics and Theoretical Physics
2523-5133
2523-5141
21
Foreword
José
Gracia-Bondía
Universidad Complutense, Zaragoza, Spain
Combinatorics
General
1
8
1
10.4171/143-1/1
http://www.ems-ph.org/doi/10.4171/143-1/1
Pre-Lie algebras and systems of Dyson–Schwinger equations
Loïc
Foissy
Centre Universitaire de la Mi-Voix, Calais, France
General
These lecture notes contain a review of the results of [15], [16], [17], and [19] about combinatorial Dyson–Schwinger equations and systems. Such an equation or system generates a subalgebra of a Connes–Kreimer Hopf algebra of decorated trees, and we shall say that the equation or the system is Hopf if the associated subalgebra is Hopf. We first give a classi cation of the Hopf combinatorial Dyson–Schwinger equations. The proof of the existence of the Hopf subalgebra uses pre-Lie structures and is different from the proof of [15] and [17]. We consider afterwards systems of Dyson-Schwinger equations. We give a description of Hopf systems, with the help of two families of special systems (quasi-cyclic and fundamental) and four operations on systems (change of variables, dilatation, extension, concatenation). We also give a few result on the dual Lie algebras. Again, the proof of the existence of these Hopf subalgebras uses pre-Lie structures and is different from the proof of [16].
9
89
1
10.4171/143-1/2
http://www.ems-ph.org/doi/10.4171/143-1/2
Five interpretations of Faà di Bruno’s formula
Alessandra
Frabetti
Université Claude Bernard Lyon 1, Villeurbanne, France
Dominique
Manchon
Université Blaise Pascal, Aubière, France
Proalgebraic groups, Hopf algebras, operads
Associative rings and algebras
Combinatorics
Group theory and generalizations
Topological groups, Lie groups
In these lectures we present five interpretations of the Faà di Bruno formula which computes the $n$-th derivative of the composition of two functions of one variable: in terms of groups, Lie algebras and Hopf algebras, in combinatorics and within operads.
91
147
1
10.4171/143-1/3
http://www.ems-ph.org/doi/10.4171/143-1/3
A Faà di Bruno Hopf algebra for analytic nonlinear feedback control systems
W. Steven
Gray
Old Dominion University, Norfolk, USA
Luis
Duffaut Espinosa
University of NSW at the Australian Defence Force Academy, Canberra, Australia
Nonlinear control systems, nonlinear operators, Hopf algebras
Systems theory; control
Associative rings and algebras
Operator theory
General
In many applications, nonlinear input-output systems are interconnected in various ways to model complex systems. If a component system is analytic, meaning it can be described in terms of a Chen–Fliess functional series expansion, then it can be represented uniquely by a formal power series over a noncommutative alphabet. System interconnections are then characterized in terms of operations on formal power series. This paper provides an introduction to this methodology with an emphasis on feedback systems, which are ubiquitous in modern technology. In this case, a Faà di Bruno type Hopf algebra is de ned for a group of integral operators, where operator composition is the group product. Using a series expansion for the antipode, an explicit formula for the generating series of the compositional inverse operator is derived. This result produces an explicit formula for the generating series of a feedback system, which had been an open problem until recently.
149
217
1
10.4171/143-1/4
http://www.ems-ph.org/doi/10.4171/143-1/4
On algebraic structures of numerical integration on vector spaces and manifolds
Alexander
Lundervold
Bergen University College, Norway
Hans
Munthe-Kaas
University of Bergen, Norway
Geometric numerical integration, Butcher series, Lie{Butcher series, combinatorial Hopf algebras
Global analysis, analysis on manifolds
Partial differential equations
General
Numerical analysis of time-integration algorithms has applied advanced algebraic techniques for more than fourty years. An explicit description of the group of characters in the Butcher–Connes–Kreimer Hopf algebra fi rst appeared in Butcher’s work on composition of integration methods in 1972. In more recent years, the analysis of structure preserving algorithms, geometric integration techniques and integration algorithms on manifolds have motivated the incorporation of other algebraic structures in numerical analysis. In this paper we will survey algebraic structures that have found applications within these areas. This includes pre-Lie structures for the geometry of flat and torsion free connections appearing in the analysis of numerical fl ows on vector spaces. The much more recent post-Lie and D-algebras appear in the analysis of flows on manifolds with flat connections with constant torsion. Dynkin and Eulerian idempotents appear in the analysis of non-autonomous flows and in backward error analysis. Non-commutative Bell polynomials and a non-commutative Faà di Bruno Hopf algebra are other examples of structures appearing naturally in the numerical analysis of integration on manifolds.
219
263
1
10.4171/143-1/5
http://www.ems-ph.org/doi/10.4171/143-1/5
Simple and contracting arborification
Emmanuel
Vieillard-Baron
Université de Bourgogne, Dijon, France
Ordinary differential operators, trees, moulds, arborification, Hopf algebras
Combinatorics
Order, lattices, ordered algebraic structures
Ordinary differential equations
General
We present a complete exposition of Ecalle’s arbori fication–coarborifi cation formalism, which is an essential component of his Mould Calculus, and we include in particular original results on the composition of arbori fied moulds. The connections with recent works regarding combinatorial Hopf algebras on trees are made but we give all the proofs in a self contained way, and we treat numerous examples.
265
353
1
10.4171/143-1/6
http://www.ems-ph.org/doi/10.4171/143-1/6
Strong QCD and Dyson–Schwinger equations
Craig
Roberts
Argonne National Laboratory, USA
confinement, dynamical chiral symmetry breaking, Dyson-Schwinger equations, hadron spectrum, hadron elastic and transition form factors, in-hadron condensates, parton distribution functions, $U_A$(1)-problem
Quantum theory
General
The real-world properties of quantum chromodynamics (QCD) – the strongly interacting piece of the Standard Model – are dominated by two emergent phenomena: con finement; namely, the theory’s elementary degrees-of-freedom – quarks and gluons – have never been detected in isolation; and dynamical chiral symmetry breaking (DCSB), which is a remarkably effective mass generating mechanism, responsible for the mass of more than 98% of visible matter in the Universe. These phenomena are not apparent in the formulae that de fine QCD, yet they play a principal role in determining Nature’s observable characteristics. Much remains to be learnt before con finement can properly be understood. On the other hand, the last decade has seen important progress in the use of relativistic quantum field theory, so that we can now explain the origin of DCSB and are beginning to demonstrate its far-reaching consequences. Dyson–Schwinger equations have played a critical role in these advances. These lecture notes provide an introduction to Dyson–Schwinger equations (DSEs), QCD and hadron physics, and illustrate the use of DSEs to predict observable phenomena.
355
458
1
10.4171/143-1/7
http://www.ems-ph.org/doi/10.4171/143-1/7
Four Faces of Number Theory
Kathrin
Bringmann
Universität zu Köln, Germany
Yann
Bugeaud
IRMA Strasbourg, France
Titus
Hilberdink
University of Reading, UK
Jürgen
Sander
Universität Hildesheim, Germany
Number theory
Computer science
Primary: 11-02, 11J81; Secondary 11A63, 11J04, 11J13, 11J68, 11J70, 11J87, 68R15
Mathematics
Number theory
Transcendence, algebraic number, Schmidt Subspace Theorem, continued fraction, digital expansion, Diophantine approximation
This book arose from courses given at the International Summer School organized in August 2012 by the number theory group of the Department of Mathematics at the University of Würzburg. It consists of four essentially self-contained chapters and presents recent research results highlighting the strong interplay between number theory and other fields of mathematics, such as combinatorics, functional analysis and graph theory. The book is addressed to (under)graduate students who wish to discover various aspects of number theory. Remarkably, it demonstrates how easily one can approach frontiers of current research in number theory by elementary and basic analytic methods. Kathrin Bringmann gives an introduction to the theory of modular forms and, in particular, so-called Mock theta-functions, a topic which had been untouched for decades but has obtained much attention in the last years. Yann Bugeaud is concerned with expansions of algebraic numbers. Here combinatorics on words and transcendence theory are combined to derive new information on the sequence of decimals of algebraic numbers and on their continued fraction expansions. Titus Hilberdink reports on a recent and rather unexpected approach to extreme values of the Riemann zeta-function by use of (multiplicative) Toeplitz matrices and functional analysis. Finally, Jürgen Sander gives an introduction to algebraic graph theory and the impact of number theoretical methods on fundamental questions about the spectra of graphs and the analogue of the Riemann hypothesis.
11
23
2015
978-3-03719-142-2
978-3-03719-642-7
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/142
http://www.ems-ph.org/doi/10.4171/142
EMS Series of Lectures in Mathematics
2523-5176
2523-5184
Lectures on Universal Teichmüller Space
Armen
Sergeev
Steklov Mathematical Institute, Moscow, Russia
Global analysis, analysis on manifolds
Differential geometry
Primary: 58B20, 58B25, 58B34; Secondary: 53C55, 53D50
Calculus + mathematical analysis
Teichmüller spaces, conformal maps, quasisymmetric homeomorphisms, Kähler manifolds, geometric quantization, noncommutative geometry
This book is based on a lecture course given by the author at the Educational Center of Steklov Mathematical Institute in 2011. It is designed for a one semester course for undergraduate students, familiar with basic differential geometry, complex and functional analysis. The universal Teichmüller space $\mathcal T$ is the quotient of the space of quasisymmetric homeomorphisms of the unit circle modulo Möbius transformations. The first part of the book is devoted to the study of geometric and analytic properties of $\mathcal T$. It is an infinite-dimensional Kähler manifold which contains all classical Teichmüller spaces of compact Riemann surfaces as complex submanifolds which explains the name “universal Teichmüller space”. Apart from classical Teichmüller spaces, $\mathcal T$ contains the space $\mathcal S$ of diffeomorphisms of the circle modulo Möbius transformations. The latter space plays an important role in the quantization of the theory of smooth strings. The quantization of $\mathcal T$ is presented in the second part of the book. In contrast with the case of diffeomorphism space $\mathcal S$, which can be quantized in frames of the conventional Dirac scheme, the quantization of $\mathcal T$ requires an absolutely different approach based on the noncommutative geometry methods. The book concludes with a list of 24 problems and exercises which can be used during the examinations.
8
12
2014
978-3-03719-141-5
978-3-03719-641-0
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/141
http://www.ems-ph.org/doi/10.4171/141
EMS Series of Lectures in Mathematics
2523-5176
2523-5184
Elliptic PDEs, Measures and Capacities
From the Poisson Equation to Nonlinear Thomas–Fermi Problems
Augusto
Ponce
Université catholique de Louvain, Belgium
Measure and integration
Potential theory
Primary: 28-02, 31-01, 35-02, 35R06; Secondary: 26B20, 26B35, 26D10, 28A12, 28A25, 28A33, 28A78, 28C05, 31B05, 31B10, 31B15, 31B20, 31B35, 35A01, 35A02, 35A08, 35A15, 35A23, 35A35, 35B05, 35B33, 35B45, 35B50, 35B51, 35B60, 35B65, 35C15, 35D30, 35J05, 35J10, 35J15, 35J20, 35J25, 35J60, 35J61, 35J86, 35J91, 35Q40, 35Q75, 35R05, 46E27, 46E30, 46E35, 49J40, 49J45, 46N20, 49S05
Calculus + mathematical analysis
Balayage method, Borel measure, Chern–Simons equation, continuous potential, diffuse measure, Dirichlet problem, elliptic PDE, Euler–Lagrange equation, extremum solution, fractional Sobolev inequality, Frostman’s lemma, Hausdorff measure, Hausdorff content, Kato’s inequality, Laplacian, Lebesgue set, Lebesgue space, Marcinkiewicz space, maximum principle, minimization problem, Morrey’s imbedding, obstacle problem, Perron’s method, Poisson equation, potential theory, precise representative, reduced measure, regularity theory, removable singularity, Riesz representation theorem, Schrödinger operator, semilinear equation, Sobolev capacity, Sobolev space, subharmonic, superharmonic, sweeping-out method, Thomas–Fermi equation, trace inequality, Weyl’s lemma
Winner of the 2014 EMS Monograph Award! Partial differential equations (PDEs) and geometric measure theory (GMT) are branches of analysis whose connections are usually not emphasized in introductory graduate courses. Yet, one cannot dissociate the notions of mass or electric charge, naturally described in terms of measures, from the physical potential they generate. Having such a principle in mind, this book illustrates the beautiful interplay between tools from PDEs and GMT in a simple and elegant way by investigating properties like existence and regularity of solutions of linear and nonlinear elliptic PDEs. Inspired by a variety of sources, from the pioneer balayage scheme of Poincaré to more recent results related to the Thomas–Fermi and the Chern–Simons models, the problems covered in this book follow an original presentation, intended to emphasize the main ideas in the proofs. Classical techniques like regularity theory, maximum principles and the method of sub- and supersolutions are adapted to the setting where merely integrability or density assumptions on the data are available. The distinguished role played by capacities and precise representatives is also explained. Other special features are: • the remarkable equivalence between Sobolev capacities and Hausdorff contents in terms of trace inequalities; • the strong approximation of measures in terms of capacities or densities, normally absent from GMT books; • the rescue of the strong maximum principle for the Schrödinger operator involving singular potentials. This book invites the reader to a trip through modern techniques in the frontier of elliptic PDEs and GMT, and is addressed to graduate students and researchers having some deep interest in analysis. Most of the chapters can be read independently, and only basic knowledge of measure theory, functional analysis and Sobolev spaces is required.
10
14
2016
978-3-03719-140-8
978-3-03719-640-3
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/140
http://www.ems-ph.org/doi/10.4171/140
EMS Tracts in Mathematics
23
Foundations of Garside Theory
Patrick
Dehornoy
Université de Caen, France
François
Digne
Université de Picardie Jules-Verne, Amiens, France
Eddy
Godelle
Université de Caen, France
Daan
Krammer
University of Warwick, Coventry, UK
Jean
Michel
Université Denis Diderot Paris 7, France
Group theory and generalizations
Combinatorics
Category theory; homological algebra
Computer science
Primary: 20Fxx, 20F05, 20F10, 20F36, 20F60, 20F65, 20M05, 20M10; Secondary: 05A05, 18B40, 18G35, 20B30, 20F55, 20L05, 20M50, 20N02, 68Q17
Groups + group theory
Group, monoid, category, greedy decomposition, normal decomposition, symmetric normal decomposition, Garside family, Garside map, Garside element, Garside monoid, Garside group, word problem, conjugacy problem, braid group, Artin–Tits group, Deligne–Luzstig variety, self-distributivity, ordered group, Yang–Baxter equation, cell decomposition
Winner of the 2014 EMS Monograph Award! This text is a monograph in algebra, with connections toward geometry and low-dimensional topology. It mainly involves groups, monoids, and categories, and aims at providing a unified treatment for those situations in which one can find distinguished decompositions by iteratively extracting a maximal fragment lying in a prescribed family. Initiated in 1969 by F. A. Garside in the case of Artin’s braid groups, this approach turned out to lead to interesting results in a number of cases, the central notion being what the authors call a Garside family. At the moment, the study is far from complete, and the purpose of this book is both to present the current state of the theory and to be an invitation for further research. There are two parts: the bases of a general theory, including many easy examples, are developed in Part A, whereas various more sophisticated examples are specifically addressed in Part B. In order to make the content accessible to a wide audience of nonspecialists, exposition is essentially self-contained and very few prerequisites are needed. In particular, it should be easy to use the current text as a textbook both for Garside theory and for the more specialized topics investigated in Part B: Artin–Tits groups, Deligne-Lusztig varieties, groups of algebraic laws, ordered groups, structure groups of set-theoretic solutions of the Yang–Baxter equation. The first part of the book can be used as the basis for a graduate or advanced undergraduate course.
6
1
2015
978-3-03719-139-2
978-3-03719-639-7
European Mathematical Society Publishing House
Zuerich, Switzerland
10.4171/139
http://www.ems-ph.org/doi/10.4171/139
EMS Tracts in Mathematics
22
Analytic Projective Geometry
Eduardo
Casas-Alvero
Universitat de Barcelona, Spain
Geometry
51-01, 51N15; 51N10, 51N20
Geometry
Projective geometry, affine geometry, Euclidean geometry, linear varieties, cross ratio, projectivities, quadrics, pencils of quadrics, correlations
Projective geometry is concerned with the properties of figures that are invariant by projecting and taking sections. It is considered one of the most beautiful parts of geometry and plays a central role because its specializations cover the whole of the affine, Euclidean and non-Euclidean geometries. The natural extension of projective geometry is projective algebraic geometry, a rich and active field of research. Regarding its applications, results and techniques of projective geometry are today intensively used in computer vision. This book contains a comprehensive presentation of projective geometry, over the real and complex number fields, and its applications to affine and Euclidean geometries. It covers central topics such as linear varieties, cross ratio, duality, projective transformations, quadrics and their classifications – projective, affine and metric –, as well as the more advanced and less usual spaces of quadrics, rational normal curves, line complexes and the classifications of collineations, pencils of quadrics and correlations. Two appendices are devoted to the projective foundations of perspective and to the projective models of plane non-Euclidean geometries. The presentation uses modern language, is based on linear algebra and provides complete proofs. Exercises are proposed at the end of each chapter; many of them are beautiful classical results. The material in this book is suitable for courses on projective geometry for undergraduate students, with a working knowledge of a standard first course on linear algebra. The text is a valuabl