Termine

Let X be a proper, smooth rigid space and G a commutative rigid group. We study the relationship between G-representations of the fundamental group of X and G-Higgs bundles on X. This is joint work with Ben Heuer and Mingjia Zhang.

A conjecture of Berger says that on a simply connected manifold all of whose geodesics are closed, all geodesics are closed with the same length. In this thesis we will prove that this conjecture holds for three dimensional manifolds. Namely we will show that the $\IS^1$-action induced by the geodesic flow is free. This is done in two parts. First we do some cohomological computations involving the Morse theory of the free loop space of $M$. Secondly we use methods from symplectic geometry and topology to investigate the topology of the quotient of the unit tangent bundle $T^1M/\IS^1$ by the $\IS^1$ action that comes from the geodesic flow. We show that this quotient admits the structure of a smooth symplectic manifold and that this symplectic manifold is symplectomorphic to $\IS^2\times \IS^2$ equipped with a split form. These two approaches lead at the end to a solution of Berger's conjecture, when the underlying manifold is three dimensional.

Cohomology of (\phi, \Gamma)-modules was studied by Herr, Liu, and Kedlaya-Pottharst-Xiao. Kedlaya-Pottharst-Xiao proved finiteness, duality, and Euler characteristic formula for cohomology of families of (\phi, \Gamma)-modules. In these consecutive talks, we will present an alternative proof of finiteness and duality using analytic geometry introduced by Clausen-Scholze and 6-functor formalism refined by Heyer-Mann. One advantage of this approach is that it applies to families over Banach Qp-algebras that are not topologically of finite type over Qp. In the first talk, we will explain solid locally analytic representations and their geometric interpretation. For later use, we will also introduce the notion of ?solid overconvergent analytic representations?. In the second talk, we will give a proof of finiteness and duality. Key ingredients include a criterion of smoothness proved by Heyer-Mann and Tate-Sen axioms.

The question of the existence of free subgroups in a given group traces back to a problem posed by von Neumann. When such subgroups exist, their genericity has been the focus of extensive research. In this talk, we formulate and investigate a version of the genericity question for subgroups of the affine group over a field of characteristic zero. Let $G$ denote the affine group over a field $F$, and let $\mu$ be a probability measure on $G \times G$. Consider the left or right random walk $(X_n, Y_n)$ on $G \times G$ driven by $\mu$. I will discuss several results on conditions under which the semigroup generated b $X_n$, $Y_n$ is free, either eventually or infinitely often. The talk is based on a work in progress with Richard Aoun.