Termine

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.

We study the properties of mean-field Gibbs measures with possibly singular interactions. We prove a law of large numbers and a quantitative central limit theorem with optimal rates in the full subcritical regime of temperatures T > Tc. Our treatment of the singular interaction borrows ideas from Nelson?s construction of the $\phi^4_2$ Euclidean quantum field theory and our ability to capture the entire subcritical regime relies on a sharp non-asymptotic large deviations-type estimate. Finally, we derive the quantitative CLT using a variant of Stein?s method. As a by-product of our results, we show that the free energy of the mean-field Coulomb gas is bounded, thus improving on the logarithmically growing bound obtained by Serfaty and Rosenzweig. This is joint work with Matías G. Delgadino (UT Austin).

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.

We consider particles interacting with a smooth mean-field potential and attempt to quantify the speed of convergence (log-Sobolev constant) of the associated Langevin dynamics in terms of the number N of particles and the strength of the interaction. Our main interest is in relating the scaling of the log-Sobolev constant as a function of N, to properties of the free energy of the model. We show that a certain notion of convexity of the free energy implies uniform-in-N bounds on the log-Sobolev constant. In some cases this convexity criterion is sharp, for instance this is the case in the Curie-Weiss model where we prove uniform bounds on the log-Sobolev constant up to the critical temperature. Our proof does not involve the dynamics. Instead, we decompose the measure describing interactions between particles with inspiration from renormalisation group arguments for lattice models, that we adapt here to a lattice-free setting in the simplest case of mean-field interactions. Our results apply more generally to non mean-field, possibly random settings, provided each particle interacts with sufficiently many others. Based on joint work with Roland Bauerschmidt and Thierry Bodineau.