Singularities and rigidity in geometric evolution equations

该项目侧重于几何流动，其中几何对象（如函数，曲面或黎曼度量）随着时间的推移而演变，其演变由以经典热方程为模型的微分方程决定。经典的热方程描述了热量随时间扩散时温度的变化。PI和合作者首先在材料科学、工程和应用数学中发现了这些方程，并在纯数学中得到了广泛的研究。这些几何流动是热方程的非线性推广，非线性效应导致新的现象，包括奇点的发展，即使从光滑的初始构型开始。理解和建模这些奇点是一个基本问题，无论是在理论上还是在应用科学。该项目更广泛的影响包括研究生指导、本科生指导、课程改革、编写研究生教材、传播、研讨会和会议组织，以及包括多个编辑委员会在内的其他社区服务。该项目研究几何流动，重点是里奇和平均曲率流动（MCF）的奇异性和刚度。平均曲率流是一种起源于材料科学的非线性抛物演化方程，在纯数学和应用数学中得到了广泛的研究。一个封闭的表面会尽可能有效地缩小面积，把自己拉紧。随着表面变小，流动收缩得更快，因此，奇点总是出现。关键是要理解奇点。函数理论所起的作用，既有连续的，又有离散的，有独特的延续。第二个主要方向是了解Ricci流中奇点的某些性质，包括何时爆点是唯一的，哪些爆点是刚性的，以及梯度收缩Ricci孤子的渐近结构。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 focuses on geometric flows, where a geometric object - such as a function, a surface, or a Riemannian metric - evolves over time with the evolution determined by a differential equation modeled on the classical heat equation. The classical heat equation describes the evolution of the temperature as heat spreads out over time. The equations that the PI and collaborators consider were first discovered in materials science, engineering and applied mathematics and are extensively studied in pure mathematics. These geometric flows are nonlinear generalizations of the heat equation and the nonlinear effects lead to new phenomena, including the development of singularities even when starting from a smooth initial configuration. Understanding and modeling these singularities is a fundamental problem, both theoretically and in applied science. The broader impact of the project includes graduate advising, undergraduate mentoring, curriculum reform, writing graduate textbooks, dissemination, seminar and conference organization, and other service to the community including multiple editorial boards. The project studies geometric flows focusing on singularities and rigidity in Ricci and mean curvature flow (MCF). Mean curvature flow is a nonlinear parabolic evolution equation that originated in materials science and has been intensely studied in pure and applied mathematics. A closed surface evolves to decrease its area as efficiently as possible, pulling itself tight. As the surface gets smaller, the flow contracts even faster and, thus, singularities always occur. The key is to understand the singularities. Function theory plays a role, both continuous and discrete, and unique continuation. A second main direction is to understand certain properties of singularities in Ricci flow, including when blowups are unique, which blowups are rigid, and the asymptotic structure of gradient shrinking Ricci solitons.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.