Singularity Formation in Geometric Flows

这个项目是关于微分几何的问题，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，引力是由时空曲率反射的。如果我们用微分方程对几何物体进行演化，使曲率以类似于散热的方式消散，那么几何物体通常可以得到改进。然而，一个重要的区别是，虽然散热是一个线性现象，但所有几何上有趣的方程都是非线性的。这个项目旨在理解这些非线性现象。特别有趣的是奇点形成的过程：也就是说，当一个表面即将破裂，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。此外，这些方程的离散版本也用于工程和计算机科学。几何演化方程最重要的例子是汉密尔顿的里奇流，它是佩雷尔曼证明庞加莱猜想和几何化猜想以及证明球体定理的关键工具。另一个重要的例子是欧几里德空间中曲面的平均曲率流。其中一个主要的概念是古代的解决方案。古老的解是可以无限向后延伸的解。它们通常在奇点形成之前作为几何流解的模型出现。因此，它们在理解奇点形成方面起着重要作用。这里的目标之一是在非坍缩假设的前提下，对所有低维的古老解进行分类。这类似于对椭圆方程的整个解进行分类的问题。PI还计划在合适的曲率限制下研究更高维度的奇点形成。在另一个方向上，PI感兴趣的是研究与最小曲面和自由边值问题有关的问题，以及偏微分方程在广义相对论中的应用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.