# Nonabelian Hodge theory and the P=W conjecture

# Nonabelian Hodge theory and the P=W conjecture

**In-Person and Online Talk**

Given a compact Riemann surface $C$, nonabelian Hodge theory (developed in the 1980's) relates topological and holomorphic structures on $C$. Namely, it gives a correspondence between complex representations of the fundamental group and holomorphic vector bundles on C, equipped with an extra structure called a Higgs field. In 2010, de Cataldo, Hausel, and Migliorini proposed a conjecture, now called P=W, which refines this relationship; roughly, it predicts that the Hodge theory of the moduli of representations of $\pi_1(C)$ is determined by the topology of the moduli space of Higgs bundles. Since then, similar phenomena have been observed in other settings.

In this talk, I will survey this circle of ideas, and discuss a recent proof of the original P=W conjecture, in joint work with Junliang Shen.