Title: The non-abelian Hodge locus

Abstract: Classical finiteness results of Arakelov and Parshin state that a fixed quasi-projective curve can only carry finitely many non-isotrivial families of smooth projective curves of fixed genus g. These results have been generalized by Faltings and Deligne for polarized variations of Hodge structure of arbitrary weight. In this talk, I will explain a further generalization which only depends on the topology of the base and not the algebraic structure, giving thus a partial answer to a question asked by Deligne. I will then explain an application proving the algebraicity of the non-abelian Hodge locus, partially solving a conjecture of Simpson. The results in this talk are joint work with Philip Engel.