Speaker: Tim Gowers (Cambridge)
Title: Partial associativity and rough approximate groups
Abstract: A binary operation o on a set X that is injective in each variable
separately is a group operation if and only if it is associative. But what
happens if all we know is that out of all triples (x,y,z) in X^3 there are c|X|^3
of them that satisfy the associative property x o (y o z) = (x o y) o z?
Elad Levi proved that if c is close enough to 1 (but independent of |X|),
then there must be a group G of order approximately equal to |X|, such that
the multiplication table of X agrees almost everywhere with that of G (in a
sense that is easy to make precise). In this talk I shall talk about a recent
result, proved jointly with Jason Long, that shows what happens in the "one
percent case” — that is, when c is a positive constant that could be quite small.