Mean curvature flow and Ricci flow

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The proposed research is at the intersection of differential geometry, partial differential equations, calculus of variations, stochastic analysis and general relativity. Specifically, the main focus is on two geometric versions of the heat equation: the evolution of surfaces by their mean curvature, and the evolution of curved spaces by Hamilton's Ricci flow. Mean curvature flow models many physical processes which involve an evolving surface, or interface. It is the most efficient way to decrease the area of surfaces and to evolve them towards optimal ones. Correspondingly, Ricci flow deforms curved spaces towards optimal shapes.******While many foundational results have been obtained on both flows, a central problem is that singularities will form in most relevant situations. For example if the geometry at the initial time looks like a dumbbell, then the neck will pinch off preventing one from continuing the flow in a smooth way. The main goal of the proposed research is to improve our understanding of the formation of singularities under mean curvature flow and Ricci flow, and to develop methods to continue the flow beyond the first singular time. This will facilitate many new applications both within and outside mathematics.******A long term goal of my research on mean curvature flow is to construct solutions with surgery for general mean convex hypersurfaces, widely generalizing the estimates and the methodology that I developed in my prior work with Bruce Kleiner. The idea of surgery is to carefully cut the surface shortly before a singularity forms and to heal it by gluing in suitable caps. In joint work with my postdocs and students I will investigate various topological applications of mean curvature flow with surgery, notably higher-dimensional Smale type conjectures about the topology of the moduli-space of embedded spheres.******A long term goal of my research on Ricci flow (mostly joint with Aaron Naber) is to develop a theory of generalized solutions that enable us to continue the flow through singularities. In a recent paper, we proved a new class of estimates for the Ricci flow that are strong enough to characterize solutions. Based on our estimates, we can provide a notion of weak solutions for the Ricci flow, which solves a longstanding open problem. Over the next 5 years we plan to develop the theory of these weak solutions. I'll also investigate several applications, in particular several geometric-analytic conjectures that have been left open after Perelman's solution of the Poincare conjecture.***The proposed research is at the forefront of modern mathematics. One of my main aims is to attract the best Canadian and international students in geometry and analysis to come to Toronto and to involve them in the research projects. I'll organize seminars, discussion groups, topics classes, summer schools and conferences. I'll disseminate the research broadly and will give many expository lectures.**