This proposal has two main topics. The first is to study measured metric spaces with nonnegative Ricci curvature in a general sense. Many of the known results about Ricci curvature for smooth spaces have extensions to nonsmooth spaces. However, there are also many results known for smooth spaces, for which the nonsmooth extension is unclear. The proposed research will explore the extent to which the curvature properties of nonsmooth spaces resemble those of smooth spaces. The second main topic is Ricci flow. The Ricci flow, introduced by R. Hamilton in the 1980's, is a way to evolve the "shape" of a space by means of its Ricci curvature. Recently, Perelman has made spectacular use of the Ricci flow to address the most important problems in three-dimensional topology. The long-time behavior of the Ricci flow is largely unknown and will be addressed in the proposal. In addition, the extension of Perelman's results to three-dimensional orbifolds will be considered.
Overall, the proposal is concerned with the idea of curvature. Historically, curvature was first considered for curves in the plane, and then for curves and surfaces in three-dimensional space. Riemann showed how to make sense of the curvature of a smooth space of arbitrary dimension. In fact, there are various notions of curvature - the sectional curvature defined by Riemann and the Ricci curvature, which is an averaging of the sectional curvature.The Einstein equation of general relativity is phrased in terms of Ricci curvature. Ever since Riemann's time, there has been interest in making sense of the curvature of nonsmooth spaces. Alexandrov gave a good notion of what it means for a nonsmooth space to have nonnegative sectional curvature. Recent work, in collaboration with Cedric Villani, has given a good notion of what it means for a nonsmooth space to have nonnegative Ricci curvature. The definition is in terms of "optimal transport", a subject which has a history in applied mathematics. Part of the research in the proposal uses "optimal transport" to study the geometry of nonsmooth spaces. Going the other way, ideas from geometry will be used to address issues in "optimal transport".
