Foreword

• Vladimir Rubtsov (Université d'Angers)

Room: Marilyn and James Simons Conference Center

A Random Matrix Model for the Entanglement Entropy of Free Fermions

• Leonid Pastur (King's College London)

Room: Marilyn and James Simons Conference Center 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.

Random Walks with Infinite Entropy - What's Amiss?

• Vadim Kaimanovich (University of Ottawa)

Room: Marilyn and James Simons Conference Center 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.

Benjamini-Schramm and Spectral Convergence of Rauzy Graphs

• Tatiana Nagnibeda (Université de Genève)

Room: Marilyn and James Simons Conference Center 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.

Remote - Variations on the Theme of GLB

• Grigorii Olshanskii (IITP, Moscow)

Room: Marilyn and James Simons Conference Center 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.

Remote - Vershik-Følner Sets and other Ideas of Vershik in Geometry and Group Theory

• Mikhail Gromov (IHES & NYU)

Room: Marilyn and James Simons Conference Center

Quantization Effects and Boundaries of Random Walks on Groups

• Andrey Malyutin (St. Petersburg State Univ.)

Room: Marilyn and James Simons Conference Center 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.

Remote - Følner Functions and Geometry of Følner Sets in Groups

• Anna Erschler (Sorbonne Université)

Room: Marilyn and James Simons Conference Center 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.

Polymer Topology Meets Fractal Dimension

• Sergei Nechaev (LPTMS Paris-Saclay)

Room: Marilyn and James Simons Conference Center 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.

Incidence Geometry and Tiled Surfaces

• Sergey Fomin (University of Michigan)

Room: Marilyn and James Simons Conference Center 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.

Remote - Branching in Planar Optimal Transport and Positive Definite Functions

• Fyodor Petrov (St. Petersburg State Univ.)

Room: Marilyn and James Simons Conference Center 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.

Intrinsic Hyperplane Arrangements

• Natalia Tsilevich (Bar Ilan University)

Room: Marilyn and James Simons Conference Center 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).

Machine Learning Methods for Cayley Graphs Path Finding and Embeddings

• Alexander Chervov (Institut Curie)

Room: Marilyn and James Simons Conference Center 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.

Kontsevich and Buchstaber Polynomials, Multiplication Kernels, and N-valued Group Laws

• Vladimir Rubtsov (Université d'Angers)

Room: Marilyn and James Simons Conference Center 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.

Remote - Limit Shapes Unlimited

• Andrei Okounkov (Princeton University)

Room: Marilyn and James Simons Conference Center 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.

Remote - Some Quick Formulas for the Volumes of and the Number of Integer Points in Higher-dimensional Polyhedra

• Alexander Barvinok (University of Michigan)

Room: Marilyn and James Simons Conference Center 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.

Remote - Geometry of Dimer Models

• Alexey Borodin (MIT)

Room: Marilyn and James Simons Conference Center 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.