Title: Product mixing in groups (Special Colloquium)

Abstract:

Let A, B, C be subsets of the special unitary group SU(n) of Haar measure ≥ e^{−n^1/3}. Then ABC = SU(n). In fact, the product abc of random elements a ~ A, b ~ B, c ~ C is equidistributed in SU(n).
This makes progress on a question that was posed independently by Gowers studying nonabelian variants of questions from additive combinatorics and settles a conjecture of physicists studying quantum communication complexity.
To prove our results we introduce a tool known as ‘hypercontractivity’ to the study of high rank compact Lie groups. We then show that it synergies with their representation theory to obtain our result.