Approximate groups and uniform spectral gaps

This lecture will be devoted to growth and expansion in finite and infinite Cayley graphs. The expander property, which is essential in many aspects of theoretical computer science, also has applications to analytic number theory. In the 2010s new combinatorial methods pioneered by Bourgain and Tao among others and based on the notion of approximate group have helped establish spectral gaps and the expander property for many Cayley graphs. I will present the state of the art on these questions and describe recent work in which Littlewood-Offord theorems for non-abelian random walks and diophantine dynamics play a role in establishing uniform spectral gaps in arbitrary linear groups.