16:00
The aim of this talk is to tell the story of Non-Abelian Hodge Theory for curves. The starting point is the space of representations of the fundamental group of a compact Riemann surface. This space can be endowed with the structure of a complex algebraic variety in three different ways, giving rise to three non-algebraically isomorphic moduli spaces called the Betti, de Rham and Dolbeault moduli spaces respectively.
After defining and outlining the construction of these three moduli spaces, I will describe the (non-algebraic) correspondences between them, collectively known as Non-Abelian Hodge Theory. Finally, we will see how the rich structure of the Dolbeault moduli space can be used to shed light on the topology of the space of representations.