Thursday, April 2, 2020 2:00 PM
Timothy Gowers (Cambridge)
 
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.