Symmetric spaces and discrete structures: Geometry, Dynamics, and Application

The study of discrete subgroups of semisimple Lie groups connects to many diverse areas in mathematics. A key tool in their investigation is the action of the discrete group on the symmetric space associated to the Lie group, and on its compactifications. There is a basic dichotomy between rank one Lie groups, when the symmetric space is of strict negative curvature, and Lie groups of higher rank, when the symmetric space is only non-positively curved. For many years, starting with Margulis celebrated superrigidity theorem, rigidity has been the overarching paradigm in higher rank. In recent years we have seen a shift of this paradigm towards a combination of flexibility and rigidity which opened up new research directions. In this talk I will describe some of the key properties of symmetric spaces, introduce the concept of Anosov representations and describe some of the recent developments, focussing of the interplay between geometry and dynamics. In the end I will discuss how the intricate geometry of symmetric spaces can be leveraged for graph embeddings in representation learning.