Nick Gill (Open University): On the Liebeck-Nikolov-Shalev conjecture
Let A be a subset of size at least 2 inside a finite simple group G. The L-N-S conjecture asserts that we can write G as a product of conjugates of A in a very efficient manner, i.e. the only restriction on the number of conjugates needed comes from the relative size of A and G.
I will describe recent work with Lifshitz, Pyber and Szabo which has resulted in a proof of this conjecture. There are two approaches: one involves the notion of “hypercontractivity”, the other is more algebraic and combinatorial. My focus will be on the latter which, amongst other things, involves understanding how multiplying sets “spread out” across conjugacy classes.
More details available on https://londonalgebra.wordpress.com/