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Singularity Formation in Geometric Flows

几何流中奇点的形成

基本信息

批准号：
1806190
负责人：
Simon Brendle
金额：
$ 21.23万
依托单位：
Columbia University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1806190&HistoricalAwards=false
关键词：
Singularity Formation Geometric Flows

项目摘要

This project is concerned with questions in differential geometry, which is the study of higher-dimensional shapes and their curvature. These objects play a central role in general relativity where gravitation is reflected by the curvature of space-time. Geometric objects can often be improved if we evolve them by a differential equation, so that the curvature dissipates in a way analogous to heat dissipation. However, one important difference is that, while heat dissipation is a linear phenomenon, all the geometrically interesting equations are nonlinear. This project is aimed at understanding these nonlinear phenomena. Of particular interest is the process of singularity formation: That is to say, what happens when a surface is about to break apart and the curvature becomes very large? Understanding these questions is an important problem within mathematics. In addition, discrete versions of these equations are used in engineering and computer science.The most important example of a geometric evolution equation is Hamilton's Ricci flow, which is a key tool in Perelman's proof of the Poincare and Geometrization conjectures, as well as in the proof of the Sphere Theorem. Another important example is the mean curvature flow for surfaces in Euclidean space. One of the main concept is that of an ancient solutions. Ancient solutions are solutions which can be extended infinitely far backward in time. They often arise as models for a solution to a geometric flow right before a singularity forms. As such, they play an important role in understanding singularity formation. One of the goals here is to classify all ancient solutions in low dimensions, subject to a noncollapsing assumption. This is analogous to the problem of classifying entire solutions to elliptic equations. The PI is also planning to study singularity formation in higher dimensions, under suitable curvature restrictions. In another direction, the PI is interested in studying problems related to minimal surfaces and free boundary value problems, as well as applications of partial differential equations to general relativity.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.

这个项目关注的是微分几何中的问题，这是高维形状及其曲率的研究。这些物体在广义相对论中扮演着核心角色，在广义相对论中，引力被时空的曲率所反映。如果我们通过微分方程来演化几何对象，那么它们通常可以得到改进，因此曲率以类似于散热的方式消散。然而，一个重要的区别是，虽然热耗散是线性现象，但所有几何上有趣的方程都是非线性的。本项目旨在了解这些非线性现象。特别令人感兴趣的是奇点形成的过程：也就是说，当一个曲面即将分裂而曲率变得非常大时会发生什么？理解这些问题是数学中的一个重要问题。几何演化方程最重要的例子是汉密尔顿的里奇流（Ricci flow），它是佩雷尔曼证明庞加莱方程和几何化方程的关键工具，也是证明球面定理的关键工具。另一个重要的例子是欧氏空间中曲面的平均曲率流。其中一个主要的概念是古解，古解是可以在时间上无限向后延伸的解，它们经常作为奇点形成之前的几何流的解的模型出现。因此，它们在理解奇点的形成过程中扮演着重要的角色，这里的目标之一是在不坍缩的假设下，对所有低维的古代解进行分类。这类似于对椭圆方程的整体解进行分类的问题。PI还计划在适当的曲率限制下研究更高维的奇异性形成。在另一个方向上，PI对研究与极小曲面和自由边值问题有关的问题感兴趣，该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值进行评估来支持和更广泛的影响审查标准。