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.
Prof. Barak Weiss's lectures have been cancelled due to the war.
3. Nicolas de Saxcé (Researcher, CNRS-Université Sorbonne Paris Nord)
Title: Lattices, subspaces and diophantine approximation
Abstract: At the beginning of the twentieth century, Minkowski introduced the space of lattices in number theory. Since then, this tool has been central in many aspects of number theory, and in particular in diophantine approximation, which is the study of the distribution of rational points on the real line, or more generally, in any real space.
The goal of this mini-course will be to explain the basic concepts of "parametric geometry of numbers", which is a way to describe diagonal orbits in the space of lattices, and to use those to study questions of Schmidt on rational approximations to linear subspaces.
