Second meeting, Tel Aviv University, 20.7.17

The joint spectrum of a set of matrices.

The joint spectral radius of a set S of matrices was introduced by Rota and Strang in the 60's and encodes the maximum asymptotic rate of spatial growth of a product of elements from S. It is intimately related to the growth of eigenvalues of products of only boundedly many elements from S by theorems of Berger-Wang and Bochi. In this talk I will present a multi-dimensional version of this notion, where one looks at the full vector of eigenvalues and is naturally led to the notion of joint spectrum of S. I will describe some of its properties, its relation to random matrix products and state some open problems. Joint work with Cagri Sert.

On a possible counter-example to the p-adic Littlewood conjecture in positive characteristic

We will discuss a possible counter-example to the p-adic Littlewood conjecture in characteristic p=3. The talk will be mainly expository and will include the statement of open problems. This is work in progress with Erez Nesharim and Fred Lunnon.

First-order rigidity for higher-rank lattices.

I'll sketch a proof for the following result: if G is a higher-rank non-uniform lattice and H is a finitely generated group that has the same first-order theory (in the sense of logic) as G, then H is isomorphic to G. Time permitting, I will talk about other aspects of the first-order theory of higher-rank lattices, such as word width. Joint work with Alex Lubotzky and Chen Meiri.

Word Measures on Classical Groups, Surfaces and their Mapping Class Group

Let w be a fixed word in F_2, the free group on two generators. Sample two independent Haar-random matrices A and B from a unitary, orthogonal or symplectic compact group, and evaluate w(A,B). Quite surprisingly, the probability measure of the random matrix w(A,B) can be basically determined from properties of w which involve surfaces and their mapping class group, or, equivalently, solutions to certain equations involving w. This is joint work with Michael Magee.

You are cordially invited.