1. Manfred Einsiedler (Professor, ETH Zurich)
Title: Totally geodesic submanifolds and arithmeticity
Abstract: In this mini course we will discuss the recent theorem of Bader-Fisher-Miller-Stover concerning arithmeticity for hyperbolic manifolds: As conjectured by Reid and McMullen a hyperbolic manifold is arithmetic if and only if there are infinitely many maximal proper totally geodesic submanifolds. This gives a striking connection between the geometry of hyperbolic manifolds and number theory. Moreover the proof uses among other things a crucial input from homogeneous dynamics. We first outline the overall proof structure and then will spend some time discussing the various ingredients.
2. Barak Weiss (Professor, Tel Aviv University)
Title: Random walks on homogeneous spaces, and Diophantine approximation on fractals
Abstract: In the last two decades, following breakthrough work of Bourgain-Furman-Lindenstrauss-Mozes and Benoist-Quint, followed by work of Eskin-Mirzakhani, there has been great progress in understanding random walks on homogeneous spaces, and on other related spaces. In work of Simmons-Weiss, Cert-Prohaska-Shi, Dayan-Ganguly-Weiss and forthcoming work of Khalil-Luethi-Weiss, this dynamical input has been used to understand number-theoretic properties of typical points on certain sufficiently regular fractals. I will survey this work, focusing on two main points:
the link between random walks, and number theory;
the main dynamical ideas for analyzing random walks. Specifically, I will give an outline of the “exponential drift” approach of Benoist and Quint in the simplest interesting case.