(joint work with Peter Scholze) In our joint work with Scholze we need to give a meaning to statements like "the stack of principal G-bundles on the curve is smooth of dimension 0" and construct "smooth perfectoid charts on it". The problem is that in the perfectoid world there is no infinitesimals and thus no Jacobian criterion that would allow us to define what is a smooth morphism. The good notion in this setting is the one of a cohomologically smooth morphism, a morphism that satisfies relative Poincaré duality. I will explain a Jacobian criterion of cohomological smoothness for moduli spaces of sections of smooth algebraic varieties over the curve that allows us to solve our problems.[-]

In 1981, Drinfeld enumerated the number of irreducible $l$-adic local systems of rank two on a projective smooth curve fixed by the Frobenius endomorphism. Interestingly, this number looks like the number of points on a variety over a finite field. Deligne proposed conjectures to extend and comprehend Drinfeld's result. By the Langlands correspondence, it is equivalent to count certain cuspidal automorphic representations over a function field. In this talk, I will present some counting results where we connect counting to the number of stable Higgs bundles using Arthur's non-invariant trace formula.[-]

Motives represent hidden building blocks for both number theory and geometry. Automorphic representations are spectral objects with the analytic power to resolve some of the deepest questions in modern harmonic analysis. It has long been thought that there were fundamental relations between these very different sides of mathematics. We shall describe conjectures on the explicit nature of some of these relations, as expressed in terms of the automorphic and motivic Galois groups. If time permits, we shall comment on how these universal groups might extend to the broader theories of mixed motives and exponential motives.[-]