Non-abelian Hodge theory and the P=W conjecture

In the first part of my talk, I will introduce the P=W conjecture by de Cataldo, Hausel, and Migliorini (2010), predicting that the perverse filtration associated with the Hitchin system is identified with the weight filtration associated with the corresponding character variety, via non-abelian Hodge theory. Then I will discuss a proof of the conjecture in joint work with Davesh Maulik, where we combine some ideas from enumerative geometry and representation theory.