To what extent does the algebraic structure of a topological group determine its topology? As an example, many (but not all) examples of real Lie groups G have a unique Lie group structure, meaning that every abstract isomorphism G -> G is necessarily continuous. In this talk, I'll discuss a recent, much stronger, result for groups of homeomorphisms of manifolds: any homomorphism from Homeo(M) to any other separable topological group is necessarily continuous.

This work is part of a broader program to show that the topology and the algebraic structure of the group of homeomorphisms (or diffeomorphisms) of a manifold are intimately linked. This circle of ideas has important consequences in geometric topology and dynamics of group actions, and I'll describe a few applications and connections during the talk.