The goal of the proposal is to study various geometrically motivated equations and their applications. The proposal will mostly center around ideas involving Ricci curvature, however aspects of it will also involve harmonic maps, spectral analysis, metric-measure spaces, and stochastic analysis. Each of these topics is somehow concerned with the bending and intrinsic structure of higher dimensional geometries. In recent years much progress has been made in each of these areas, but there are a great deal of unknown questions which remain, the solutions of which would have applications in many branches of mathematics and physics. In all there are three parts to the proposal with nine projects and seven coauthors. Each project will discuss first progress expected to be made over the next year, and then goals past that.
The first part of the proposal centers on studying the connections between Ricci curvature and the infinite dimensional analysis on path space. Recent breakthroughs have allowed researchers to realize that the two are intimately connected, and the hope is that further understanding in this area should open up new areas of research as well as solve many questions involving spaces with bounded Ricci curvature. There are two projects in this part of the proposal. The second part of the paper focuses on the regularity of manifolds with lower and bounded Ricci curvature. In particular, together with several co-authors the projects include the proving of new epsilon-regularity theorems for Einstein manifolds and showing that metric-measure spaces with lower Ricci curvature bounds are rectifiable. There are three projects in the second part of the proposal. The final part of the proposal includes four projects from various areas of geometric analysis. This includes a project in the very classical topic of elliptic equations on Euclidean space. It is surprising, but there are still many open questions in this area. In particular, the project focuses on the study of critical sets of such equations under very rough coefficients.