Singularities and rigidity in geometric evolution equations

几何演化方程中的奇异性和刚性

• 批准号: 2304684
• 负责人: William Minicozzi
• 金额: $ 40万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2304684&HistoricalAwards=false
• 关键词: Singularities; rigidity; geometric; evolution; equations

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.

这个项目的重点是几何流，其中一个几何对象-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确定。经典的热方程描述了温度随时间的变化。PI和合作者考虑的方程首先在材料科学，工程和应用数学中发现，并在纯数学中得到广泛研究。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，即使从光滑的初始配置开始。理解和模拟这些奇点是一个基本的问题，无论是在理论上还是在应用科学中。该项目的更广泛影响包括研究生咨询，本科生指导，课程改革，编写研究生教科书，传播，研讨会和会议组织，以及对社区的其他服务，包括多个编辑委员会。该项目研究几何流，重点是里奇和平均曲率流（MCF）中的奇异性和刚性。 平均曲率流是一个非线性抛物型发展方程，起源于材料科学，在理论数学和应用数学中得到了广泛的研究。一个封闭的曲面会尽可能有效地缩小其面积，并将自身拉紧。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。函数论起着作用，既有连续的，又有离散的，而且唯一的连续性。第二个主要方向是理解Ricci流中奇点的某些性质，包括何时爆破是唯一的，哪些爆破是刚性的，以及梯度收缩Ricci孤子的渐近结构。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。