Geometry, Topology, and Applications

In 1872 Felix Klein introduced a new approach to view geometry as the study of symmetries. This approach has been very influential both in mathematics as well as in physics. It allows to study different geometric structures in a unified framework, and to investigate the interplay between topological manifolds and their geometric realizations in a new way.

In the past twenty years several new developments at the intersection of geometry and topology led to a revival of these ideas and paradigm shifts in the study of geometric structures.

In this talk I will highlight some of these developments. In the end I will hint at the potential of these ideas for the discovery of structure in data.