Jason Long, Squarepoint Capital

Abstract

Suppose that a binary operation * on a finite set X is injective in each coordinate and is also associative. It is easily shown that (X,*) is a group. But what happens if one only knows that * is associative a positive proportion of the time? Other results in additive combinatorics might lead us to expect there to be an underlying 'group-like' structure that is responsible for the large number of associative triples, and we prove that this is indeed the case. The heart of the proof is purely combinatorial, since many of the relevant algebraic properties can be translated into statements about counting certain subgraphs of a particular hypergraph. This is joint work with Tim Gowers.