Title: Non-abelian Hodge correspondence and the P=W conjecture
Speaker: Zili Zhang
Abstract: Fix a complex projective curve C and a reductive group G. There are three moduli spaces with the pair (C,G): the character variety M_B, the moduli of flat connections M_dR, and the moduli of Higgs bundles M_Dol. The non-abelian Hodge correspondence says there are natural homeomorphisms among the three moduli spaces, and hence identify the cohomology groups of them. The geometric structures of the moduli spaces induce various filtrations in the cohomology groups. De Cataldo-Hausel-Migliorini conjectured in 2012 that the Perverse filtration (P) of M_Dol is identical to the Hodge-theoretic weight filtration (W) of M_B; the P=W conjecture. We will introduce the background and recent progress of the nonabelian Hodge correspondence and the P=W conjecture. The talk is not aimed at specialists.