Vorträge in der Woche 09.11.2020 bis 15.11.2020

Donnerstag, 12.11.2020: Die (tropische) Kontsevichformel und ihre Verallgemeinerung auf mehrere crossratios

Christoph Goldner ( Universität Tübingen)

Donnerstag, 12.11.2020: Level set methods for scalar curvature in dimension three

Dr. Daniel Stern (University of Toronto)

I'll discuss some new tools for studying the influence of scalar curvature on 3-manifolds, exploiting a relationship between scalar curvature and the level sets of harmonic functions and S^1-valued harmonic maps. These methods share features with the well-known minimal surface and inverse mean curvature flow techniques, while yielding some estimates reminiscent of those arising from Dirac operator methods. We'll describe applications to the study of the Thurston norm of closed 3-manifolds, and the ADM mass of asymptotically flat 3-manifolds. (Based in part on joint work with Hugh Bray, Demetre Kazaras, and Marcus Khuri.)

Freitag, 13.11.2020: Hypersurfaces moving by concave functions of curvature

Stephen Lynch

Evolution processes that deform geometric objects by their curvature arise naturally in many contexts, and have been used to solve difficult problems lying at the intersection of analysis, geometry and topology. Here we discuss the evolution of embedded hypersurfaces in a Euclidean or Riemannian background space with normal speed given by a concave function of the principal curvatures. Our focus is on the application of PDE methods to obtain insights into the singularities formed by such flows in case the initial embedding is highly non-convex. In particular, we find new flows that preserve natural curvature conditions in a Riemannian background space, and whose almost-singular regions are modeled on convex ancient solutions. We expect these results to play a role in future efforts to understand the interplay between curvature and the topology of hypersurfaces.