# Groups and Dynamics Seminar

## 2019 - 2020

RLM 10.176, Friday from 2 to 3 pm

10/11 Rostislav Grigorchuk.

Title: Spectra of graphs and groups.

Abstract: First I will give a short account of what is known about spectrum of discrete Laplace operator on infinite regular graphs and groups.
Then I will discuss the question of Valette and Fujiwara "Can one hear the shape of a group?"
And finally, using the lamplighter group I will explain that the spectrum of the group can have infinitely many gaps.
The relation of the spectral theory of graphs and groups with other topics, in particular with L^2-Betti numbers and random Schrodinger operators, also will be mention if the time permit.

09/27

Peter Burton.

Title: Free harmonic analysis and hyperbolic geometry.

09/20

09/13 John Chaika. TBA

Title: A prime system with many and big self-joinings

Abstract: Let (X,mu,T) be a measure preserving system. A factor is a
system (Y,nu,S) so that there exists F with SF=FT and so that F pushes mu
forward to nu. A measurable dynamical system is prime if it has no
non-trivial factors. A classical way to prove a system is prime is to show
it has few self-joinings, that is, few T x T invariant measures that on X
x X that project to mu. We show that there exists a prime transformation
that has many self-joinings which are also large. In particular, its
ergodic self-joinings are dense in its self- joinings and it has a
self-joining that is not a distal extension of itself. As a consequence we
show that being quasi-distal is a meager property in the set of measure
preserving transformations, which answers a question of Danilenko. This is
joint work with Bryna Kra.

09/06 Anthony Quas.

Title: Lyapunov spectral stability and collapse for random composition of Blaschke Products

Abstract: We consider random dynamical systems where at each stage one applies a Blaschke product.
Motivated by questions on mixing of invariant measures on the unit circle, we study the composition
of the associated Perron-Frobenius operators, acting on a Hardy space. We describe the complete
Lyapunov spectrum, and address questions of stability of the spectrum under perturbations to
the operator cocycle.

## Summer 2019
Room on the 11th floor, Monday 10 am to 2 pm

08/26

08/19 Max Chaudkhari.

Title: Superharmonic functions and Northshield's criterion of amenability.

Abstract: We will discuss an extension of Northshield's criterion of amenability to group actions and its applications. In particular, we will describe a new approach to the study of amenability of Thompson's group $F$. The talk is based on a recent preprint by Maksym Chornyi: arxiv