To what extent does the algebraic structure of a topological group
determine its topology? 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, stronger result for groups of homeomorphisms of
manifolds: every homomorphism from Homeo(M) to any other separable
topological group is necessarily continuous.

This is part of a broader program to show that the topology and
algebraic structure of the group of homeomorphisms (or
diffeomorphisms) of a manifold M are closely linked, and also deeply
connected to the topology of M itself. Time permitting, we'll
discuss applications in geometric topology, groups acting on
manifolds, and connections with a new program to study the
quasi-isometry type of homeomorphism groups.