Autumn Semester 2021
Equidistribution of random walks in tori and applications to super-strong approximation, Weikun He, 22-24-25 November.
Monday: 9-10:30 AM , Room: Y21-D-68a, UZH
Wednesday: 9-10:30AM , Room: Y27-H-35/36, UZH
Thursday: 9-10:30 AM , Room: Y34-F-01, UZH
Abstract: Given a subgroup of SL(d,Z), we can look at its reduction modulo arbitrary integers. If we fix a generating set, we obtain a family of corresponding Cayley-Schreier graphs. We can ask whether this family is a family of expander graphs. This property is also known as expansion or spectral gap or super-approximation.
Recently, in a joint work with Nicolas de Saxcé, we proved that the answer is yes if the identity component of the Zariski closure of the group is absolutely simple. The proof is mainly based on the work of Salehi-Golsefidy-Varju, the work of Bourgain-Varju and an improvement of the Bourgain-Furman-Lindenstrauss-Mozes theorem about the equidistribution of random walks in tori.
The goal of this mini-course is to give an introduction to this topic and convey ideas of the main steps of the proof.
Lecture 1: Definition of super-approximation and Bourgain-Gamburd expansion machine.
Lecture 2: Equidistribution of random walks on the torus and its implication.
Lecture 3: Expansion in low levels and how to put things together.
Spring Semester 2022
Around Margulis Normal Subgroup Theorem, Uri Bader, 24-25 May 2022.
Tuesday 15-17 ETH ML F39
Wednesday 10-12 ETH NO C 44
Abstract: I will start with presenting a proof of Margulis NST and then discuss various generalizations.
I will relate the above to the dynamics of a group on its space of positive definite functions and the related space of characters.
I will present a variety of new results which are proved using (non-commutative) ergodic theoretical techniques.