Math 201A: Random walk on a compact group.

Winter 2020

Lectures: T-Th 9:00 - 10:25  APM 7321
Office Hour: Send me an e-mail.

Send me an e-mail, 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 Mallahi-Karai:

  • Fourier analysis on compact groups: the Peter-Weyl theorem and the Plancherel theorem.
  • Gowers's quasi-randomness.
  • 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, Breuillard-Green-Tao, Pyber-Szabo, de Saxce.
  • Spectral gap; Kazhdan's property (T).
  • Expanders: Cheeger constant.
  • Spectral gap in a single scale: the Bourgain-Gamburd machine.
  • Survey on super-approximation: works of Bourgain-Varju, Varju-SG, SG, SG-Zhang, SG-Longo, He-de Saxce.
  • Spectral gap in a multi-scale setting: Littlewood-Payley decomposition in locally random groups.
  • Spectral independence of two compact groups.
  • Related open problems and projects.


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


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 Publication-Vol(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: