On the de Rham-Hitchin system

The de Rham-Hitchin system is a deformation of the Hitchin system in prime characteristic. It plays a main role in the prime characteristic Non Abelian Hodge Theory, relating Higgs bundles and connections. In this talk, I will survey what I know about the geometry and topology of the de Rham-Hitchin system, emphasizing how it relates to complex geometry and number theory. We start by motivating some correspondences on the category level using p-curvatures. Then we do some moduli theory to get a correspondence at the moduli space level. Finally, we deduce some consequences on the cohomologies of the moduli spaces.