ACTION NOW WANDERING SEMINAR

First meeting, Ben Gurion University, 27.5.21

Let Gamma_g be the fundamental group of the closed orientable surface of genus g. Fix an element gamma in Gamma_g and let f: Gamma_g -> S_N be a uniformly random homomorphism to the symmetric group S_N.

We develop new techniques to study the random permutation f(gamma), and derive several results which are analogous to well-known results when Gamma_g is replaced with a free group. This is joint work with Michael Magee.

On some fixed point theorems and a question of Kazhdan and Yom Din

In 1962 Ryll-Nardzewski announced a new fixed point theorem and showed how one can deduce from it the existence of an invariant mean-value on the Banach space of weakly almost periodic functions on a topological group. In the first part of my talk I will discuss various aspects of this old theorem and provide some related new fixed point theorems. Studying a question of Kazhdan and Yom Din, regarding linear actions of a discrete countable group G on a dual Banach space V^*, I will show in the second part of the talk that while in some cases their proposed fixed point theorem holds, the answer to their question in its full generality is negative.

A Delone set in Euclidean space is a point set that is on the one hand uniformly separated, and on the other hand, intersects every ball of large enough radius. Delone sets Y and Z are BD-equivalent if there exists a bounded displacement (BD) mapping between them, namely a bijection f: Y -> Z such that the quantity ||f(y)-y|| (y in Y) is bounded. We study the cardinality of the set of equivalence classes and show that a minimal space of Delone sets either contains a set which is BD to a lattice, in which cases all sets in the space are such, or there are continuum many BD-classes represented in the space. If time permits, we will discuss some applications of this dichotomy to several interestin constructions from the theory of aperiodic order. Based on a joint work with Yotam Smilansky.

We will discuss joint work with Nir Avni which gives some evidence for the following conjecture: there is a first order sentence P in the language of groups, such that for every finitely generated group Delta, the sentence P holds in Delta if and only if Delta is an irreducible lattice in a higher rank semisimple group.

To a word w in a free group F on r generators, and a group G, one can associate a word map w: G^r -> G. When G is compact, such a word map induces a natural probability measure on G, and one can study the corresponding random walk. We consider the collection of random walks on SL_n(Z/p^kZ) induced by w, as p, k and n vary. Surprisingly, it turns out that various mixing properties of these random walks have equivalent algebro-geometric charcterizations in terms of the singularity properties of the algebraic maps w: SL_n(C)^r -> SL_n(C) and its concatenations (also called convolutions) w * w * ... * w: SL_n(C)^r -> SL_n(C). We explain this connection, and further show that word maps in semisimple Lie groups and Lie algebras have nice singularity properties after sufficiently many self-convultions (with bounds depending only on the word). As a consequence, we obtain uniform results on the above collection of random walks.

Based on a joint work with Yotam Hendel, see arXiv:1912.12556

Ehud Ettun, Daniel Schwarzwald and Tsachik Gelander.

The concert will be held in room -101.