Abstract

The aim of this talk is to give a gentle introduction to (arithmetic and geometric applications of ) homogeneous dynamics. In particular, I will focus on the study of joinings in homogeneous dynamics. 

As a warm-up, I will start with two simple but "artificial" arithmetic applications of dynamical systems and explain how these systems can be "joined" together. Then, I will discuss joint equidistribution results of classical objects (such as points on spheres , rational subspaces and their shapes of lattices, complex-multiplication elliptic curves) and their relation to joinings on homogeneous spaces. This is related to joint works with Einsiedler, Luethi, Michel, Shapira, Wieser.

Our analysis of rational planes in four space and their shapes of lattices turned out to be related to Seifert surfaces. Finally, I will discuss a recent joint work with Feller, Miller and Wieser which classifies the pairs of binary quadratic forms that arise as the Seifert forms of pairs of disjoint Seifert surfaces of genus one. In particular, we use Gauss composition to systematically generate pairs of Seifert surfaces that are non-isotopic in the 4-ball.