Iulian Ion Simion
Babeş-Bolyai University, Cluj-Napoca, Romania

By the product expansion of two subsets $A$ and $B$ of a group $G$, we mean a way of measuring how much $A\cdot B$ grows with respect to the size of $A$ and the size of $B$. Such questions are related to a conjecture of Thompson, which states that for any non-abelian finite simple group there is a conjugacy class $C$ of $G$ such that $C^2=G$. Motivation also comes from the theory of expander graphs. If $S$ is a generating set for $G$ and $S^m=G$, for some $m$ then $2m$ is an upper bound on the diameter of the Cayley graph $\Gamma=\Gamma(G,S)$. This in turn translates into bounds on the expansion constant of $\Gamma$. After presenting more background and further motivating conjectures, we comment on recent contributions to covering numbers of normal subsets in simple algebraic groups.