Seminar Series by Prof. Wilderich Tuschmann

Part 1: Spaces and moduli spaces of (Riemannian) metrics

Part 2: Quantifying molecular similarity via Gromov-Hausdorff distances

Speaker: Prof. Wilderich Tuschmann

Department of Mathematics, Karlsruhe Institute of Technology, Germany

Date: Thursday, 13 February 2025

Time: 10:30 AM. -12:30 PM.

Room: M303

Abstract Part 1: Consider a smooth manifold with a Riemannian metric satisfying some sort of geometric constraint like, for example, positive scalar curvature, non-negative Ricci or negative sectional curvature, being Einstein, Kähler, Sasakian, of special holonomy, etc. A natural question to ponder is then what the space of all such metrics does look like. Moreover, one can also study this question for the corresponding moduli spaces of metrics,i.e., quotients of the former by the diffeomorphism group of the manifold, acting by pulling back metrics. These spaces are customarily equipped with the topology of smooth convergence on compact subsets and the quotient topology, respectively, and their topological properties then provide the right means to measure ‘how many’ different metrics and geometries the given manifold actually does exhibit. The history of the subject as a whole indeed goes back more than a century, and since H. Weyl’s early result on the connectedness of the space of positive Gaussian curvature metrics on the two-sphere and the foundings of Teichmüller, infinite- dimensional manifold and Lie group theory, uniformization and geometrization, the study of spaces of metrics and their corresponding moduli has been a topic of interest for differential geometers, global and geometric analysts and topologists alike. In my talk, I will provide a gentle introduction to the subject with a focus on lower curvature bounds and present recent results and open questions about the global topological properties of moduli spaces of nonnegatively curved Riemannian metrics on manifolds, and as well, if time permits, also corresponding facts and questions for the moduli of nonnegative curvature metrics on RCD spaces.