EXC 2044 - T05: Curvature, shape and global analysis

Basic data for this project

Description

Riemannian manifolds or geodesic metric spaces of finite or infinite dimension occur in many areas of mathematics. We are interested in the interplay between their local geometry and global topological and analytical properties, which in general are strongly intertwined. For instance, it is well known that certain positivity assumptions on the curvature tensor (a local geometric object) imply topological obstructions of the underlying manifold. Curvature bounds also determine certain properties of elliptic operators (or other analytic objects), which sometimes can be related to topological properties of the underlying manifold via index-theoretic methods. The curvature or the local geometry is determined by geodesics, whose analytic properties are thus in close relation to analytical (and topological) properties of the underlying space. However, on infinite-dimensional (shape) spaces, existence of geodesics is non-trivial since they are governed by PDEs (and not ODEs as in finite dimensions).