Math 207A: Super-approximation

Spring 2022

 Lectures: T-Th 11:00 - 12:20 APM 5402 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:

Unlike my previous topics courses, this course will be a more survey-style course with an emphasis on the main ideas rather than details of many proofs. The main concepts and more fundamental results will be discussed more thoroughly, but in order to cover all the main results on this growing topic, we skip the details of many of the proofs. As the title says, in this course, we will learn about super-approximation : the study of random-walks on congruence quotients of subgroups of $$S$$-arithmetic groups. We will mention some of many applications of this topic to other parts of mathematics: construction of expanders, affine sieve, sieve in groups, hyperbolic groups, Apollonian packing, Zaremba's conjecture, etc. Here are some of the articles that will be discussed in this course:

• J. Bourgain, A. Gamburd, Uniform expansion bounds for Cayley graphs of $$\mathrm{SL}_2(\mathbb{F}_p)$$, Annals of Mathematics 167 (2008) 625–642.
• P. Varju, Expansion in $$\mathrm{SL}_d(\mathcal{O}_K/I)$$, $$I$$ square-free, Journal of European Mathematical Society 14, no. 1, (2012) 273–305.
• A. S. Golsefidy and P. Varju, Expansions in perfect groups, Geometric And Functional Analysis 22, no. 6, (2012) 1832–1891.
• A. S. Golsefidy, Super-Approximation, I: $$p$$-adic semisimple case, International Mathematics Research Notices 2017, no 23, (2017) 7190-7263.
• A. S. Golsefidy, Sum-product phenomena: the $$\mathfrak{p}$$-adic case, Journal d'Analyse Mathematique 142 (2020) 349--419.
• J. Bourgain, P. Varju, Expansion in $$\mathrm{SL}_d(\mathbb{Z}/q\mathbb{Z})$$, $$q$$ arbitrary, Inventiones Mathematicae 188, no 1, (2012) 151-173.
• J. Bourgain, A. Furman, E. Lindenstrauss and S. Mozes, Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus, J. Amer. Math. Soc. 24 (2011), 231-280.
• de Saxce and W. He, Linear random walks on the torus, Duke Math. J.

Notes related to lectures and supplementary materials: I will share some of my notes here.
• Here is my lecture notes so far.