NOTE: unusual day and location
(joint with FRG Workshop on Derived Categories and Moduli Spaces)

Abstract: Given a compact Riemann surface $C$, nonabelian Hodge theory relates
topological and algebro-geometric objects associated to $C$. Specifically,
complex representations of the fundamental group $\pi_1(C)$ are in
correspondence with algebraic vector bundles on $C$, equipped with an extra
structure called a Higgs field. This gives a transcendental matching between two
very different moduli spaces for $C$: the character variety of $\pi_1(C)$
(parametrizing its representations) and the so-called Hitchin moduli space of
$C$ (parametrizing vector bundles with Higgs field).

In 2010, de Cataldo, Hausel, and Migliorini proposed a conjectural relation --
now called the P=W conjecture -- between these two spaces. This conjecture
gives a precise link between the topology of the Hitchin space and the Hodge
theory of the character variety, imposing surprising constraints on each side.
In the first part of this talk, I will give an introduction to this circle of ideas; in
the second part, I will survey some recent progress towards understanding this
conjecture, using ideas from compact hyperkahler geometry and geometry in
characteristic $p$.