Award: DMS 1207654, Principal Investigator: John W. Lott
The principal investigator will work on three projects in differential geometry and analysis on manifolds. These involve both linear and nonlinear aspects of geometric analysis. The first project is the long-time behavior of the Ricci flow in three and four dimensions. In three dimensions, the main open problems are the finiteness of the number of surgeries, the decay of the sectional curvature and the long-time asymptotics of the geometry. In four dimensions, the goal is to use Ricci flow to identify canonical geometries. The second project concerns optimal transport. We will work on a construction of parallel transport on the space of probability measures, equipped with the Wasserstein metric. We will also work on isoperimetric inequalities and sharp Sobolev inequalities for metric-measure spaces with positive Ricci curvature. The third project will develop an index theorem for transverse Dirac-type operators on manifolds with Riemannian foliations. The principal investigator will also develop a model of differential K-theory based on infinite-dimensional vector bundles.
Optimal transport is the study of the optimal way to transport mass. It was first considered by Monge in 1781. In the past few decades, links have been established between optimal transport, partial differential equations and applied mathematics. More recently, links have also been found with differential geometry. It seems likely that ideas from geometry will find applications to optimal transport. The Ricci flow has clear importance for geometry and topology. The methods that are being proposed to study its long-term behavior, in particular the use of monotonic quantities adapted to collapsing flows, may well have application to other geometric flows.
