This proposal deals with curved spaces of higher dimension. An interesting question is how to measure the curvature of a nonsmooth space. In the case of sectional curvature, this was initiated in the 1930's by Alexandrov, who gave a good notion of what it means for a singular space to have nonnegative sectional curvature. Recent work by the proposer, joint with Cedric Villani, together with related work by Karl-Theodor Sturm, has given a good notion of what it means for a singular space to have nonnegative Ricci curvature. The proposed research will explore properties of spaces with nonnegative Ricci curvature or, more generally, with Ricci curvature bounded below. In a different direction, the Ricci curvature gives a way to smooth out the geometry of a space, by means of the Ricci flow. This flow was introduced by Richard Hamilton in the 1980's, who used it to characterize the topology of three-dimensional smooth spaces with nonnegative Ricci curvature. Recently, Perelman has proved the biggest conjectures in three-dimensional topology, namely the Poincare Conjecture and Thurston's Geometrization Conjecture, using Ricci flow. Despite Perelman's great achievements, there are many open questions concerning the Ricci flow in dimension three and in higher dimensions. The proposed research will address some of these questions. Another way that singular spaces arise is when a higher-dimensional space is foliated into lower dimensional spaces. The parametrizing space for such a foliation is almost always topologically singular. Part of the proposed research is to do analysis on such spaces, or more precisely to prove a transverse index theorem.
There are various notions of curvature, which coincide in the traditional setting of two-dimensional surfaces in three-dimensional space. For a higher dimensional smooth space, not necessarily living in a flat space, these different notions are called the sectional curvature, the Ricci curvature and the scalar curvature. Each one is an averaging of the previous one, i.e. the Ricci curvature is an averaging of sectional curvature and the scalar curvature is an averaging of Ricci curvature. The Ricci curvature enters in physics through Einstein's equations of general relativity. Part of the work described above concerns ``optimal transport''. This is the study of the optimal way to transport mass in a curved space. It was initiated by Monge in the 1780's and has had a revival in recent decades, with application to partial differential equations and applied mathematics. We have shown that it also has application in differential geometry. Conversely, concepts from differential geometry have application to optimal transport.
This project was concerned with various aspects of differential geometry. Differential geometry is the study of curved spaces in arbitrary dimension. The goal is to understand how curvature affects the shape of an object. One dynamical approach to this is the Ricci flow, introduced by Richard Hamilton, which uses curvature to deform the shape of an object. The project looked at the dynamics, under the Ricci flow, of various types of spaces. These included unbounded spaces (quasiprojective manifolds) and singular spaces (orbifolds). A way that curvature affects the global shape of objects is through index theory, following the work of Atiyah and Singer. The index as originally defined by Atiyah and Singer is an integer. They later constructed indices that take value in "K-theory". Since then, other indices have been defined which carry more refined information. In one recent example, the index takes in "differential K-theory". The latter combines the topological information from K-theory and the geometric information from differential forms. During the time of the project, an index theorem was proven in which the indices take value in differential K-theory; this theorem subsumes most of the work in local index theory of the past thirty years. Finally, other directions were explored in the project, such as mean curvature flow in a Ricci flow background, and the indices of transverse differential operators.
