Geometry, Dynamics, and PDEs in Riemannian and Finsler Spaces

Abstract

Award: DMS 1510611, Principal Investigator: Dmitri Burago

This research project takes its origin at the concept of length of curves and goes on to develop familiar and important ideas for physical or dynamical systems under limited hypotheses. Among those ideas are notions of stability, instability, and entropy that address the variations of larger geometric properties as a metric is subjected to small perturbations. The project has a component related to discrete forms of geometric data and the Laplace operator on metric measure spaces. Two graduate students of the principal investigator will be working on experimental measurements of frequencies of vibrating bodies of unusual shapes.

It is known that the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a Riemannian manifold are approximated by eigenvalues and eigenvectors of a (suitably weighted) graph Laplace operator of a proximity graph on an epsilon-net. One of these projects seeks to extend that result to the Laplacian on differential forms and to get better estimates for the Laplacian on metric measure spaces, including stability results. It would be very interesting to understand which spaces with bounded geometry can be approximated to an additive error by graphs with uniform bounds on degrees of vertices and lengths of edges. So far, we can do that only for the 2-plane and not even for 3-space, but we do not know a single counterexample to the main conjecture in this area. One project aims to prove Michel's Conjecture on boundary rigidity under reasonable hypotheses. Another project will study dynamical systems with positive entropy in which entropy is generated locally, that is, positive entropy is generated in arbitrarily small tubes around one trajectory. Another goal is to prove ellipticity of the Busemann surface area.