Let C be a compact Riemann surface, and G=\pi_1(C) its fundamental group. While enumerating the representations of G over finite fields, Hausel and Rodrigues-Villegas have noticed that they always obtained palindromic polynomials. One way to interpret this symmetry is to say that the cohomology of the variety M_B of representations of G over complex numbers (so called character variety) admits a "curious" Poincaré duality. Similar dualities were previously studied by de Cataldo and Migliorini, but their theory could only be applied to the variety M_D of Higgs bundles on C, which is homeomorphic to M_B, but not isomorphic as an algebraic variety. Matching up these two dualities (the "curious" one being conjectural) boils down to an equality of two filtrations (perverse and weight) on H^*(M_B) of very different nature. This equality became known as P=W conjecture. In my talk, I will explain this story in more detail, and give some indication of how this conjecture was eventually proved. Based on joint work with T. Hausel, A. Mellit, O. Schiffmann.

Afternoon tea will be provided at 3:30pm