Dynamics and Singularities of Geometric Flows

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The project focuses on understanding fundamental properties of nonlinear heat equations in geometry. The laws of nature are often expressed as differential equations and these fall into several broad categories. The classical heat equation is one prototype; it describes the evolution of temperature in space over time. This and similar equations play a central role in diverse fields, such as physics, economics, information theory, engineering and mathematics. One of the fundamental principles of the heat equation is diffusion: heat spreads out over time as the temperature becomes more and more constant. In the classical heat equation, the heat particles move independently and do not interact. When the particles interact with each other, then the situation becomes much more complicated and the evolution is governed by a nonlinear partial differential equation. This is seen in many geometric problems, such as the evolution of interfaces where the governing principle is a nonlinear version of the heat equation. The equations describing the evolution of interfaces is known as mean curvature flow; it was initially studied in material science and now plays a role in many areas of science, engineering and mathematics. In addition to the research component of this proposal the project also will also train students: the PI is currently advising several PhD students as well as working with an advanced undergraduate student. The PI is writing books on research topics of current interest based on graduate courses (on the heat equation and on the mean curvature flow) with his colleague Toby Colding. The PI serves on the editorial board of eight highly regarded mathematics journals, including the Annals of Mathematics. The PI is also working on curriculum reform at his home institution, the Massachusetts Institute of Technology.The project attacks fundamental questions on geometric flows, including properties of singularities and dynamical properties of the flow. In mean curvature flow (MCF), a sub-manifold evolves over time to decrease its area as efficiently as possible, pulling itself tight. Over time, the non-linearity dominates and singularities develop. The key is to understand these singularities. Most of the progress has been for hyper-surfaces where the flow can be expressed as a single equation. In higher co-dimension, the flow is a complicated system of interacting equations and much less is known. This project focuses on fundamental questions in higher co-dimension MCF: 1. Which singularities are generic and which can be perturbed away?2. What are the dynamics of the flow near a singularity?3. What does the flow look like near a singularity? Is the blow up unique?4. When are singularities isolated?These are absolutely fundamental questions and progress will have significant implications. One of the key tools is the subtle interplay between function theory and geometry. Many of the techniques developed apply to a broader class of systems, including Ricci flow.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.