Singular Ricci flow, Einstein flow and index theory

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

The PI proposes to work on three projects in differential geometry and analysis on manifolds. These involve both linear and nonlinear aspects of geometric analysis. Geometric flows, such as those addressed in this proposal, have applications to many branches of science, including materials science and cosmology. The research in this proposal will enhance the theoretical understanding of such flows. The Ricci flow is fundamental in modern geometry. Most of the work on Ricci flow has been done on smooth Ricci flows or Ricci flows with surgery. On the other hand, many partial differential equations have a well-defined class of singular solutions. The proposed work will explore a remarkable class of singular solutions for the three-dimensional Ricci flow. Similarly, for many partial differential equations, there are good notions of weak solutions. Understanding limits of Ricci flow with surgery may shed light on the problem of developing a good notion of weak Ricci flow. The Einstein flow is a geometric flow with very different features than parabolic flows. The proposed research will elucidate some general features of the Einstein flow. The long-time behavior of vacuum spacetimes is of evident interest in cosmology. Finding a framework for differential K-theory based on infinite-dimensional cocycles will lead to new directions in local index theory and the associated functional analysis. It may also have topological consequences. There are three main topics in this proposal.(1) Singular Ricci flows. In joint work with Kleiner, the PI has shown that there is a limit of Perelman's Ricci flow with surgery, as the surgery parameter goes to zero. The PI proposes to study refined properties of such limits, and more generally of singular Ricci flows.(2) Einstein flow. The Einstein flow is a flow which describes the long-time behavior of a vacuum spacetime that has a foliation by constant mean curvature spatial hypersurfaces. The PI will use techniques from collapsing theory in order to understand the geometry of the Einstein flow in the collapsing case.(3) Differential K-theory. Differential K-theory is a topological theory that combines K-theory with differential forms. In joint work with Gorokhovsky, the PI will construct infinite-dimensional models of differential K-theory.