Large scale index, positive scalar curvature and manifold topology

• Applicant: Professor Dr. Thomas Schick
• Subject Area: Mathematics
• Term: from 2016 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 321324296

Given a smooth compact manifold M without boundary we study the question which kinds of geometric structure M accommodates. Of special importance is ---motivated by cosmology--- the scalar curvature, and we study in particular the space of Riemannian metrics of positive scalar curvature on M. A long-standing question is: when is this space non-empty (i.e.~when does M admit a metric of positive scalar curvature)? More generally: what is the topology of this space? For example, what are its homotopy groups?A fundamental differential equation of quantum physics, the Dirac equation for spinors, is closely connected to positive scalarcurvature, as already observed by Schr\"odinger. Modern refinements, of which the research program is a part, use operator algebras and their K-theory to unveil the subtle global information about geometry which is encoded in this relation.More specifically, large scale index theory of the Dirac operator is a rather new and very successful tool to study positive scalar curvature questions. It uses operator algebras adapted to non-compact Riemannian manifolds, where the focus is on thelarge-scale properties. This has seen tremendous development in recent years. The theory connects to long-standing and deep questions in operator algebras and topology, like the Baum-Connes conjecture and the Novikov conjecture. In the project, we will develop further the paradigm of large scale index theory. We will identify new geometric situations where this paradigm applies, and we will make new geometric-topological constructions which allow to apply the method for the classical case of a closed manifold. This will require the development of fine local analytic methods. These will be combined in new ways with sophisticated tools from homotopy theory and homology.

• DFG Programme: Research Grants