Topology of character varieties for surface groups

Spaces of isomorphism classes of homomorphisms (so-called character varieties) of the fundamental group of a surface to a Lie group G play a role in several areas of mathematics, and are also interesting in their own right. We shall outline how the theory of Higgs bundles provides analytical tools for studying these spaces and, in particular, their topology, and we shall present some recent results on the case when G is the isometry group of a classical Hermitian symmetric space of non-compact type, obtained in joint work with Bradlow, Garcia-Prada and Mundet i Riera.