Abstract：

For a complex projective curve C and a reductive group G, the character variety M_B and the moduli of Higgs bundles M_Dol are canonically homeomorphic via the Simpson's correspondence and hence the cohomology groups of them are naturally identified. 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 some background and recent progress of the P=W conjecture.