Lecture series

Charles Bordenave

Strong convergence of random matrices : In 2005 Haagerup and Thorbjørnsen have established the convergence in operator norm of any non-commutative polynomial in independent Gaussian Hermitian matrices as their common dimension grows to infinity. This form of convergence of matrix algebras generated here by random matrices has attracted since then a lot of attention. It has always powerful consequences. In these lecture, I will present a strategy based on combinatorics and probability to establish this form of convergence.

Nir Lazarovich and Arie Levit

Stability and surface groups : We all seek stability in life — for group theorists, this means that every approximate action is approximately an action. Stability has received a lot of attention recently, in part because of its connections to soficity and to Connes’ embedding problem. The goal of the lecture series is to present a proof that surface groups are flexibly stable. The proof involves different tools from hyperbolic geometry and combinatorial group theory.
Based on joint work with Y. Minsky.

Michael Magee and Doron Puder

The asymptotic statistics of random covering surfaces : Let Γ be the fundamental group of a closed connected orientable surface of genus g>1. We develop a new method for integrating over the space of homomorphisms Hom(Γ,S_N), where S_N is the symmetric group. Equivalently, this is the space of all vertex-labeled, N-sheeted covering spaces of the closed surface of genus g. In particular, we study the local statistics of the random permutation obtained as the image of a fixed γ∈Γ through a random homomorphism to S_N. Our methods lie in representation theory of S_N as well as in combinatorial topology. We plan to convey our main results, some ideas from the technique, and some open questions.

Research talks

Maria Gerasimova

Stability in permutations of some non-amenable groups : Stability in permutations of some non-amenable groups", "We will discuss some constructions which give new examples of non-amenable flexibly stable in permutations groups. The main question we consider is the following: when are amalgamated products and HNN-extensions over finite groups stable in permutations?. This question is closely related to the properties of the restrictions of actions from a group to a finite subgroup. I will present some partial answers to this question and discuss further development of this topic.

Anna Roig Sanchis

Random hyperbolic 3-manifolds : We are interested in studying the behavior of geometric invariants of hyperbolic 3-manifolds, such as the length of their geodesics. A way to do so is by using probabilistic methods. That is, we consider a set of hyperbolic manifolds, put a probability measure on it, and ask what is the probability that a random manifold has a certain property. There are several models of construction of random manifolds. In this talk, I will explain one of the principal probabilistic models for 3 dimensions and I will discuss the length spectrum–the set of lengths of all closed geodesics–of a 3-manifold. Moreover, by encoding this invariant through the function N_L = {number of closed geodesics of a fixed length L>0}, I will show that for non-compact random hyperbolic 3-manifolds, N_L behaves asymptotically like a Poisson-distributed random variable.

Matan Seidel

Measures induced by words on GL_n(q) and free group algebras : Fix a finite field K and a word w in a free group F. A w-random element in GL_n(K) is obtained by substituting the letters of w with uniform random elements of GL_n(K) . For example, if w=abab⁻², a w-random element is ghgh⁻² with g,h independent and uniformly random in GL_n(K) . The moments of w-random elements reveal a surprising structure which relates to the free group algebra K[F]. In this talk I will describe what we know about this structure and draw some analogies to w-random permutations.

Itamar Vigdorovich

Spectral gap properties of characters, and vanishing in higher rank lattices : Consider a center-free higher rank lattice with property (T). Any sequence of ergodic invariant random subgroups converges to the trivial IRS - this may be viewed as a baby version of the main seven samurai result. We show an analogous statement for characters: and sequence of distinct characters must converge pointwise to the dirac character. The proof shares some analogies with the IRS case, but requires a study of spectral gap properties associated with characters. We obtain similar results for congruence characters.
In the talk, I'll define the notion of characters, draw the analogies with the IRS case, and explain the main ideas of our proof. It is based on a joint work with Arie Levit and Raz Slutsky

Tomer Zimhoni

Random Permutations from Free Products : Let Γ=G₁*...*G_r be a free product of groups such that every G_i is either a finite group or a copy of Z. Every γ∈Γ induces a probability measure on S_N by choosing uniformly at random a homomorphism φ∈Hom(Γ, S_nN) and considering φ(γ). My talk will focus on some results relating to the expected number of fixed points of such a random permutation. These results reflect some algebraic properties of γ and are stated and proven in a topological perspective.
This is a joint work with Professor Doron Puder.