Cohomology rings of character varieties

Abstract

In this talk, I give an introduction and present some recent progress towards understanding the cohomology rings of character varieties of Riemann surfaces, such as a proof of the $P=W$ conjecture and the computation of the zero-dimensional cohomological Hall algebra (COHA). In the case of punctured sphere, I present an explicit description relating the cohomology rings to the Hilbert scheme of ${\mathbb{C}}^2$, refining conjectures of Hausel–Letellier–Rodriguez-Villegas and Chuang–Diaconescu–Donagi–Pantev. I explain how the general case should be related to the symplectic geometry of the Hilbert scheme.