Math 201A: Random walk on a compact group.

Winter 2020
Lectures:

TTh

9:00  10:25

APM 7321

Office Hour:

Send me an email.



Send me an email, we can meet and discuss math (possibly in a coffee shop).

Course description:
The following is a tentative list of topics to be covered; Local randomness and spectral independence are from a joint work in progress with Amir Mohammadi and Keivan MallahiKarai:
 Fourier analysis on compact groups: the PeterWeyl theorem and the Plancherel theorem.
 Gowers's quasirandomness.
 Locally random groups: definition, basic examples, basic properties, an important mixing inequality.
 Product results for subsets with large metric entropy in locally random groups.
 Survey on product results for subsets with positive metric entropy: works of Helfgott, BreuillardGreenTao, PyberSzabo, de Saxce.
 Spectral gap; Kazhdan's property (T).
 Expanders: Cheeger constant.
 Spectral gap in a single scale: the BourgainGamburd machine.
 Survey on superapproximation: works of BourgainVarju, VarjuSG, SG, SGZhang, SGLongo, Hede Saxce.
 Spectral gap in a multiscale setting: LittlewoodPayley decomposition in locally random groups.
 Spectral independence of two compact groups.
 Related open problems and projects.

Prerequisite:
will be kept to a minimum; but a first course on functional analysis and graduate group theory would be useful.

Resources:
I will not follow a particular book, but I will post the related books and articles in the course's webpage.
Here are a few related references:
 E. M. Stein, Topics in harmonic analysis.
 B. Bekka, P. de la Harpe, A. Valette, Kazhdan's property (T).
 Terry Tao's notes on Expansion in groups.
 Emmanuel Kowalski's lecture notes on Expander graphs.
 Emmanuel Breuillard's lecture notes on Approximate subgroups,
and notes on Expander graphs, property(tau), and approximate subgroups.
 Alex Lubotzky's survey article on Expander graphs in pure and applied mathematics.
 MSRI PublicationVol(61) Thin groups and superstrong approximation.
 Hoory, Linial, and Wigderson's article on Expander graphs and their
applications is an excellent survey on the applications of Expanders in theoretical computer science.
We will not discuss these applications in the course.

Notes related to lectures and supplementary materials:
