In this course, we will discuss the following topics:

• Expander graphs: Cheeger constant, spectral gap;
• Property(T) and Property(tau);
• Margulis's explicit construction of expander graphs;
• Selberg's 3/16-theorem;
• Product Theorem in finite simple groups of Lie type: Helfgott's SL(2,P), some of the ideas of the general case;
• When finitely many matrices can give us expanders: Bourgain-Gamburd result(s), Varju's result(s), some of the ideas of the general case;
• Some applications: affine sieve, sieve in groups;

Basics of group theory (Math200A is more than enough) and linear algebra (a good undergrad-level course) will be assumed. My primary goal is to convey the main ideas in this growing subject. So though I will mention the best known results, it would be more than enough if you understand how the proofs work for two-by-two matrices.

Here are some lecture notes:

• 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.

• Helfgott's articles on Product theorems for SL(2,P) and SL(3,P).
• Larsen and Pink's article on Finite subgroups of algebraic groups.
• Breuillard, Green, and Tao's artcile on Approximate subgroups of linear groups.
• Pyber and Szabo's article article on Growth in finite simple groups of Lie type.
• Bourgain and Gamburd's article on Uniform expansion bounds for Cayley graphs of SL(2,P).
• Bourgain, Gamburd, and Sarnak's article on Affine linear sieve, expanders, and sum-product.
• Varju's article on Expansion in SL(d,O/I), I square-free.
• Bourgain and Varju's article on Expansion in SL(Z/qZ), q arbitrary.
• Salehi Golsefidy and Varju's article on Expansion in perfect groups.
• Salehi Golsefidy and Sarnak's article on Affine sieve.

• The letter grades will be based on the homework assignments and attendance.