DescriptionThe question of introducing suitable curvature notions on non-smooth spaces is very challenging. While curvature is a very natural notion in the context of surfaces and higher dimensional smooth curved spaces, called Riemannian manifolds, there is no "canonical way" to introduce curvature concepts on singular spaces like, for example, networks or general metric spaces. On the other hand, curvature notions may be introduced by way of exploiting phenomena appearing under negative or positive curvature bounds. These phenomena offer then potential guidelines to choose curvature definitions for these spaces and to provide the motivation for these choices.Here are some examples: In the case of simply connected spaces, negative curvature implies exponential growth of balls or thinness of triangles, there are close relations between lower positive curvature bounds and smallest positive eigenvalue bounds of the Laplace operator on compact spaces, and lower positive curvature bounds imply particular intersection properties of balls. Last but not least, there is the famous Gauss-Bonnet Theorem relating Gaussian curvature to a topological invariant, the Euler characteristic. The aim of this project is to investigate various choices of curvature definitions on singular spaces and, possibly, to make comparisons between them. This area is at the interface between countless mathematical disciplines: combinatorial curvatures interacting with Algebraic Topology and Computational/Geometric Group Theory (group theory enters though the concept of Cayley graphs and complexes and leads to challenging questions like the famous Word Problem); global negative curvature notions (Alexandrov curvature or Gromov hyperbolicity) interacting with Asymptotic Geometry (like properties of naturally defined boundaries at infinity), researchers even studied hyperbolicity properties of the World Wide Web; or introduction of various lower Ricci curvature bounds using concepts of the Laplace operator from Spectral Theory (Bakry-Emery curvature) or Optimal Transport (Ollivier Ricci curvature, Erbar/Maass entropic curvature, curvature-dimension inequalities a la Sturm or Lott-Villani for geodesic metric measure spaces). The literature for this topic is vast, so I can only list a few sources. Students choosing this topic should have taken the 3H course Differential Geometry to be familiar with classical curvature notions for surfaces. CorequisiteThe course Riemannian Geometry IV is recommended.ResourcesAs mentioned before, the literature is vast. Here are some sources:
email: N Peyerimhoff |