Abstract

In the past two decades there has been intense interest in products of subsets in finite groups.
Two important examples are Gowers' theory of Quasi Random Groups and its applications by
Nikolov, Pyber, Babai and others, and the theory of Approximate Subgroups and the Product Theorem
of Breuillard-Green-Tao and Pyber-Szabo on growth in finite simple groups of Lie type of bounded rank,
extending Helfgott's work. These deep theories yield strong results on products of three subsets
(covering, growth). What can be said about products of two subsets?

I will discuss a recent joint work with Michael Larsen and Pham Tiep on this challenging problem,
focusing on products of two normal subsets of finite simple groups, and deriving some applications.
The proofs involve algebraic geometry, representation theory and combinatorics.