Representations, Probability, and Beyond: A Journey into Anatoly Vershik's World

18 nov. 2024, 09:00 → 19 nov. 2024, 18:00 Europe/Paris

Marilyn and James Simons Conference Center (IHES)

Representations, Probability, and Beyond: A Journey into Anatoly Vershik's World 
Workshop in memory of A.M. Vershik

November 18-19, 2024, Mikhail Gromov (IHES & NYU), Sergei Nechaev (LPTMS Paris-Saclay) and Volodya Rubtsov (Univ. Angers) organize a two-day workshop devoted to the memory of Anatoly Vershik, who passed away earlier this year.

Anatoly Vershik (1933-2024) was a Russian mathematician who made important contributions in several fields of mathematics. In particular, he is renowned for his joint work with Sergei V. Kerov on the theory of representations of infinite symmetric groups and on applications of the longest strictly increasing subsuite problem in group theory. 

 

 

Invited Speakers:

• Alexander BARVINOK (Univ. of Michigan)
• Alexey BORODIN (MIT)
• Alexander CHERVOV (Institut Curie)
• Anna ERSCHLER (Sorbonne Univ.)
• Sergey FOMIN (Univ. of Michigan)
• Mikhail GROMOV (IHES & NYU)
• Vadim KAIMANOVICH (Univ. of Ottawa)
• Andrey MALYUTIN (St. Petersburg State Univ.)
• Tatiana NAGNIBEDA (Univ. Genève)
• Sergei NECHAEV (LPTMS Paris-Saclay)
• Andrey OKOUNKOV (Princeton Univ.)
• Grigorii OLSHANSKII (IITP, Moscow)
• Leonid PASTUR (King's College London)
• Fyodor PETROV (St. Petersburg State Univ.)
• Volodya RUBTSOV (Univ. d'Angers)
• Natalia TSILEVICH (Bar Ilan University)

 

Conference-A.Vershik.pdf

lundi 18 novembre

09:30 → 09:50 Coffee, opening

09:50 → 10:00 Foreword

Orateur: M. Vladimir Rubtsov (Université d'Angers)

Vidéo

10:00 → 10:45 A Random Matrix Model for the Entanglement Entropy of Free Fermions

Quantum entanglement, a special form of quantum correlation, is an important ingredient of modern quantum mechanics and related fields. Much of the extensive literature on entanglement considers quantum correlations between a particular subsystem (a block) and the rest of the system (environment), and uses the entanglement entropy as a quantifier of entanglement. It is assumed that the system size N is much larger than the block size L, which may also sufficiently large, i.e., heuristically, 1 ≪ L ≲ N. A widely accepted mathematical version of this inequality is the regime of successive limits: first the macroscopic limit N → ∞, and then an asymptotic analysis of the entanglement entropy for L → ∞. We consider another version of the above heuristic inequality: the regime of asymptotically proportional L and N, i.e., simultaneous limits N → ∞, L → ∞, L/N → c > 0. Specifically, we deal with a quantum system of free fermions that are in their ground state and have a large random matrix as a one-body Hamiltonian. We show that the entanglement entropy obeys the volume law known for systems having a local one-body Hamiltonian but described either by a mixed state or by a pure but highly excited state.

Orateur: Leonid Pastur (King's College London)

Vidéo

10:45 → 11:30 Random Walks with Infinite Entropy - What's Amiss?

I will discuss the qualitatively new properties of random walks on groups that arise in the situation when the entropy of the step distribution is infinite.

Orateur: M. Vadim Kaimanovich (University of Ottawa)

Vidéo

11:30 → 12:15 Benjamini-Schramm and Spectral Convergence of Rauzy Graphs

Many dynamical systems admit a natural symbolic representation as subshifts over a finite alphabet. Their complexity can be studied by associating to the subshift an infinite family of finite graphs that describe the local structure of its orbits; such graphs are known as Rauzy graphs. In this talk we will be interested in convergence properties of these sequences of graphs and of their spectra, and how their limits are related to the invariant measures of the dynamical system.

Orateur: Mme Tatiana Nagnibeda (Université de Genève)

Vidéo

12:15 → 13:45 Lunch

13:45 → 14:30 Remote - Variations on the Theme of GLB

Sergei Kerov and Anatoly Vershik discovered and started to study a remarkable group, which is an infinite-dimensional analog of GL(n) over a finite field. This group, denoted by GLB, is locally compact, totally disconnected, and it bears some resemblance with reductive p-adic groups. As demonstrated in a paper by Vadim Gorin, Sergei Kerov, and Anatoly Vershik, the group GLB possesses a rich representation theory. I will describe some results and open problems related to GLB and other similar groups, partly inspired by Kirillov's orbit method. Based on joint work with Cesar Cuenca.

Orateur: M. Grigorii Olshanskii (IITP, Moscow)

Vidéo

14:30 → 14:50 Remote - Vershik-Følner Sets and other Ideas of Vershik in Geometry and Group Theory

Orateur: M. Mikhail Gromov (IHES & NYU)

Vidéo

14:50 → 15:35 Quantization Effects and Boundaries of Random Walks on Groups

We will discuss random walks on discrete groups and group boundaries, which were among A.M.Vershik's favorite research topics. In particular, we discuss the Poincaré rotation number integer quantization effect for random walks on groups acting on the circle, and how this quantization effect can be interpreted in terms of boundary theory.

Orateur: M. Andrey Malyutin (St. Petersburg State Univ.)

Vidéo

15:35 → 16:05 Coffe break

16:05 → 16:50 Remote - Følner Functions and Geometry of Følner Sets in Groups

The definition of Følner function of groups is given in 1973 paper of A.M. Vershik, his appendix to the Russian translation of Greenleaf's book on amenable groups. We give of survey of the results about this asymptotic invariant of groups, properties of Følner sets in groups and discuss open questions in this domain.

Orateur: Mme Anna Erschler (Sorbonne Université)

Vidéo

16:50 → 17:35 Polymer Topology Meets Fractal Dimension

We investigate statistical and topological properties of fractal Brownian motion with short-range interactions. The attention is paid to statistical properties of conformations with the fractal dimension Df ≥ 2 in the three-dimensional space. Using a combination of analytic arguments and Monte Carlo simulations we show that, with the increase of the fractal dimension, Df >2, typical conformations become less knotted. Our study is motivated by an attempt to mimic the statistics of unknotted polymer rings, which are known to equilibrate into the compact hierarchical structure with Df = 3 at large scales. Replacing topologically stabilized conformation by a path with the fractal dimension Df = 3, we tremendously simplify the problem since we wash out the topological constraints from the consideration.

Orateur: M. Sergei Nechaev (LPTMS Paris-Saclay)

Vidéo

mardi 19 novembre

10:00 → 10:45 Incidence Geometry and Tiled Surfaces

We show that various classical theorems of linear incidence geometry, such as the theorems of Pappus, Desargues, Möbius, and so on, can be interpreted as special cases of a general result that involves a triangulation of a closed oriented surface, or a tiling of such a surface by quadrilateral tiles. This yields a general mechanism for producing new incidence theorems and generalizing the known ones.

Orateur: M. Sergey Fomin (University of Michigan)

Vidéo

10:45 → 11:30 Remote - Branching in Planar Optimal Transport and Positive Definite Functions

Let us have two finitely supported probability distributions $\mu, \nu$ on the Euclidean plane, and the cost of transferring of mass $m$ per unit distance is proportional to $m^p$, $0 < p < 1$. Then the optimal transferring of $\mu$ to $\nu$ (the so called Gilbert -- Steiner problem) may have branching points. I want to speak about recent joint result with Danila Cherkashin that the degree of branching points must be equal to 3. The proof relies on the theory of positive definite functions, and so, unites two seemingly far topics, both beloved by Anatoly Moiseevich.

Orateur: M. Fyodor Petrov (St. Petersburg State Univ.)

Vidéo

11:30 → 12:15 Intrinsic Hyperplane Arrangements

For every irreducible complex representation of a symmetric group, we construct, in a canonical way, a so-called intrinsic hyperplane arrangement in the corresponding space. It is a direct generalization of the classical braid arrangement, has a natural description in terms of invariant subspaces of Young subgroups, and enjoys a number of remarkable properties. (Joint work with A.Vershik and S.Yuzvinsky).

Orateur: Mme Natalia Tsilevich (Bar Ilan University)

Vidéo

12:15 → 13:45 Lunch

13:45 → 14:30 Machine Learning Methods for Cayley Graphs Path Finding and Embeddings

We present the application of machine learning and reinforcement learning methods to the analysis of Cayley graphs, specifically focusing on path finding and graph embeddings. This approach is inspired by DeepMind's AlphaGo system. It is already able to overcome GAP (classical computer algebra system): we are finding paths on groups of orders 10ˆ40-10ˆ70, while GAP encounters computational limits for such large groups. The method is general and can be applied to any (finite) permutation or matrix group. Lengths of paths produced by our general approach are shorter than obtained by algorithmic and other solvers which can handle only specific groups like Rubik's Cube group. More generally we will argue that Cayley graphs provide an excellent framework for the mathematical understanding of the key concepts of modern machine learning and reinforcement learning in particular. If time permits we will describe potential applications of that technique to biological questions like construction of embeddings for proteins and drugs like small molecules.

Orateur: M. Alexander Chervov (Institut Curie)

Vidéo

14:30 → 15:15 Kontsevich and Buchstaber Polynomials, Multiplication Kernels, and N-valued Group Laws

We shall discuss several partial results of ongoing work in collaboration (with I. Gaiur & D. Van Straten and with V. Buchstaber & I. Gaiur) on
interesting properties of multiplication kernels, which include the famous Sonine –Gegenbauer formulas, examples of polynomials for
Buchstaber–Kontsevich discriminant locus given as addition laws for special 2-valued formal groups (Buchstaber–Novikov–Veselov) as well as
a connection of N-Bessel kernel discriminant loci with N-vlaued multiplication laws given by Buchstaber-Rees polynomials.

Orateur: M. Vladimir Rubtsov (Université d'Angers)

Vidéo

15:15 → 16:00 Remote - Limit Shapes Unlimited

This will be a nontechnical talk about one of AM's favorite objects in mathematics, its universal importance, and how it is continued to be used by many different researchers around the globe.

Orateur: M. Andrei Okounkov (Princeton University)

Vidéo

16:00 → 16:30 Coffee break

16:30 → 17:15 Remote - Some Quick Formulas for the Volumes of and the Number of Integer Points in Higher-dimensional Polyhedra

In this talk, based on joint works with J.A. Hartigan and with M. Rudelson, I discuss some computationally efficient formulas, motivated by the maximum entropy principle from statistical physics, to approximate the volume of and count integer points in some combinatorially interesting families of higher-dimensional polyhedra.

Orateur: M. Alexander Barvinok (University of Michigan)

Vidéo

17:15 → 18:00 Remote - Geometry of Dimer Models

Random dimer coverings of large planar graphs are known to exhibit unusual and visually apparent asymptotic phenomena that include formation of frozen regions and various phases in the unfrozen ones. For a specific family of subgraphs of the (periodically weighted) square lattice known as the Aztec diamonds, the asymptotic behavior of dimers admits a precise description in terms of geometry of underlying Riemann surfaces. The goal of the talk is to explain how the surface structure manifests itself through the statistics of dimers. Based on joint works with T. Berggren and M. Duits.

Orateur: M. Alexey Borodin (MIT)

Vidéo