Arbeitsgemeinschaft: Geometry and Representation Theory around the P=W Conjecture

Abstract

Given a smooth projective curve $C$, nonabelian Hodge theory gives a diffeomorphism between two different moduli spaces associated to $C$. The first is the moduli space of Higgs bundles on $C$ of rank $n$, which is equipped with the structure of an algebraic completely integrable Hamiltonian system. The second is the character variety of representations of the fundamental group of $C$ into $GL(n)$. In 2012, de Cataldo, Hausel, and Migliorini proposed the $P=W$ conjecture which identifies the perverse filtration on the cohomology of the Higgs moduli space with the weight filtration on the cohomology of the character variety. Recently, in 2022, two independent proofs of the $P=W$ Conjecture appeared, in work of Maulik & Shen and Hausel, Mellit, Minets & Schiffmann. The aim of the Arbeitsgemeinschaft was to understand the $P=W$ Conjecture and these two recent proofs.