The $P=W$ conjecture, first proposed by de Cataldo-Hausel-Migliorini in 2010, gives a link between the topology of the moduli space of Higgs bundles on a curve and the Hodge theory of the corresponding character variety, using non-abelian Hodge theory. In this talk, I will explain this circle of ideas and discuss a recent proof of the conjecture for $GL_n$ (joint with Junliang Shen).
For more information see https://personal.math.ubc.ca/~jbryan/Zoominar-UBC-ETH/