In this course, we will discuss the following topics:

• Random-walk on a space via a group action and finitely many elements of the group.
• Spectral gap, contraction factor, and Lyapunov exponent. Groups with property(T).
• Expanders and explicit constructions.
• Property(T) and bounded generation.
• Fourier analysis of finite groups. Mackey theory.
• Quasi-random groups and product of large subsets.
• Bourgain-Gamburd method.
• The Brueillard-Green-Tao and Pyber-Szabo theorems on approximate subgroups.
• Ping-pong argument and Nori's theorem.
• Multi-scale setting: sum-product and its application.
• Random-walk on group extensions (bounded scale case).

Here you can find my lecture notes. These notes include extra topics and problem sets. If you would like to get A+, you should do a good portion of these problems.

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.

• Emmanuel Breuillard and Alex Lubotzky's survey
• 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 \({\rm SL}(2,\mathbb{F}_p)\) and \({\rm SL}(3,\mathbb{F}_p)\).
• 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 \({\rm SL}(2,\mathbb{F}_p)\).
• Golsefidy's article on random-walk on compact semisimple \(p\)-adic analytic groups.
• Golsefidy's article on random-walk on compact perfect \(p\)-adic analytic groups.
• Golsefidy and Srivatsa's article on random-walk on \({\rm SL}_2(\mathbb{F}_p)\times {\rm SL}_2(\mathbb{F}_p)\).
• Golsefidy and Srivatsa's article on random-walk on group extensions.