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Abstract: This talk will discuss regularity results for harmonic maps into a certain kind of metric space, called Euclidean buildings.  This will include a discussion of what these spaces are, the behavior of harmonic maps into these spaces, and the connection to superrigidity results for lattices.  We will discuss previously known regularity results before presenting a recent theorem of Breiner, Mese, and myself which generalizes these considerably. 

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.