Title: Hypercontractivity and product mixing in compact Lie groups 

Abstract for the lecture series by Noam Lifshitz and Guy Kindler:
The study of geometric properties of Cayley graphs associated with non-abelian groups is a rich area of research with wide ranging applications. Here, key questions concern properties such as expansion, mixing, diameter, etc. While remarkable progress was made in this area by applying deep results in character theory, such methods seem to be limited to the case of normal Cayley graphs (those generated by unions of conjugacy classes).

In this mini-course we explore a recent approach which seems to sidestep such limitations. Inspired by techniques originally introduced in the study of the (abelian) Boolean Cube, we use a synergy between hypercontractive inequalities and dimensional lower bounds from representation theory to make progress on various problems. In particular, we will present a significant improvement of a bound of Gowers on the measure of product-free subsets in the unitary group SU(n).