Abstract: In this talk, I will survey the P=W conjecture, which describes certain structures of the cohomology of the moduli space of Higgs bundles on a curve in terms of the character variety of the curve.  I will then explain how certain symmetries of this cohomology, which are predictions of this conjecture, can be constructed using techniques from non-abelian Hodge theory in positive characteristic.  Based on joint work with Mark de Cataldo, Junliang Shen, and Siqing Zhang.