Dynamics and Singularities of Geometric Flows

该项目的重点是了解几何中非线性热方程的基本性质。自然规律通常被表示为微分方程式，这些方程可分为几大类。经典热方程是一个原型，它描述了空间中温度随时间的演变。这个方程和类似的方程在物理、经济、信息理论、工程和数学等不同领域中发挥着核心作用。热量方程的基本原理之一是扩散：随着温度变得越来越恒定，热量会随着时间的推移而扩散。在经典热方程中，热粒子独立运动，不相互作用。当粒子相互作用时，情况变得复杂得多，演化过程由非线性偏微分方程式控制。这在许多几何问题中都可以看到，例如界面的演化，其中的主导原理是热方程的非线性版本。描述界面演化的方程被称为平均曲率流；它最初是在材料科学中研究的，现在在许多科学、工程和数学领域发挥着作用。除了这项提案的研究部分，该项目还将培训学生：PI目前正在为几名博士生提供建议，并与一名高级本科生合作。这位PI正在与他的同事托比·科尔丁一起撰写关于当前感兴趣的研究主题的书籍，这些主题基于研究生课程(关于热方程和平均曲率流)。国际数学联合会是包括《数学年鉴》在内的八家颇具声望的数学期刊的编辑委员会成员。这位PI还在他的家乡麻省理工学院进行课程改革。该项目研究几何流动的基本问题，包括奇点的性质和流动的动力学性质。在平均曲率流(MCF)中，子流形随着时间的推移而演变，以尽可能有效地减小其面积，并将自己拉紧。随着时间的推移，非线性占主导地位，奇点不断发展。关键是要理解这些奇点。大多数进展都是针对超曲面的，在超曲面上，流动可以用一个方程来表示。在更高的余维中，流动是一个复杂的相互作用的方程系统，我们知道的要少得多。本项目主要研究高维MCF的基本问题：1.哪些奇点是一般的，哪些可以被扰动？2.奇点附近的流动是什么动力学？3.奇点附近的流动是什么样子的？爆炸是独一无二的吗？4.奇点什么时候是孤立的？这些绝对是基本的问题，进展将具有重大意义。其中一个关键工具是函数论和几何学之间微妙的相互作用。许多开发的技术适用于更广泛的系统类别，包括RICCI FLOW。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.