Dynamics and Singularities of Geometric Flows

几何流的动力学和奇点

基本信息

批准号：
2005345
负责人：
William Minicozzi
金额：
$ 58.95万
依托单位：
Massachusetts Institute of Technology
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-01 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005345&HistoricalAwards=false
关键词：
Dynamics Singularities Geometric Flows

项目摘要

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.

该项目着重于理解几何学中非线性热方程的基本特性。自然定律通常表示为微分方程，这些方程式属于几个广泛的类别。经典热方程是一个原型。它描述了空间随时间的变化的演变。这个和类似方程在各种领域（例如物理，经济学，信息理论，工程和数学）中起着核心作用。热方程式的基本原理之一是扩散：随着温度变得越来越恒定，热量会随着时间的流逝而扩散。在经典的热方程式中，热颗粒独立移动并且不相互作用。当粒子相互相互作用时，情况就会变得更加复杂，并且进化受非线性偏微分方程的控制。这在许多几何问题中都可以看出，例如，管理原理是热方程的非线性版本的界面的演变。描述界面演化的方程称为平均曲率流。它最初是在材料科学领域进行的，现在在许多科学，工程和数学领域都发挥了作用。除了该提案的研究部分外，该项目还将培训学生：PI目前正在为几位博士生提供建议，并与高级本科生合作。 PI正在与他的同事Toby Colding一起撰写有关当前兴趣研究主题的书籍。 PI在编辑委员会中由八个备受推崇的数学期刊（包括数学年鉴）组成。 PI还在他的家庭机构，马萨诸塞州理工学院进行课程改革。该项目攻击了几何流量的基本问题，包括奇异性的属性和流量的动态性能。在平均曲率流（MCF）中，一个子序列会随着时间的流逝而演变，以尽可能有效地降低其面积，从而使自己紧绷。随着时间的流逝，非线性占主导地位和奇异性。关键是要了解这些奇异性。大多数进度都是针对可以将流量表示为单个方程式的超曲面。在较高的共同度中，流是一个复杂的相互作用方程系统，而知之甚少。该项目重点介绍了更高的共同维度MCF：1。哪些奇异性是通用的，哪些可以扰乱的？2。奇点附近流动的动力学是什么？3。奇异性附近的流程看起来像什么？爆炸是独一无二的吗？4。什么时候孤立的奇点？这些绝对是根本的问题，进步将产生重大影响。关键工具之一是函数理论和几何形状之间的微妙相互作用。开发的许多技术都适用于包括RICCI流动在内的更广泛的系统。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响标准，被认为值得通过评估来获得支持。