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.

该项目的重点是几何流动，其中几何对象（例如函数，表面或riemannian指标）随着时间的推移而演变，而通过在经典热方程式上模拟的微分方程确定的进化。经典的热方程描述了随着热量随着时间的流逝而扩散的，温度的演变。 PI和合作者认为的方程式是在材料科学，工程和应用数学中首次发现的，并在纯数学中进行了广泛的研究。这些几何流量是热方程的非线性概括，非线性效应导致了新现象，包括从平滑的初始构型开始时就会发展奇点。理解和建模这些奇异性是理论上和应用科学中的一个基本问题。该项目的更广泛影响包括研究生咨询，本科指导，课程改革，撰写研究生教科书，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。该项目研究了几何流量，重点是RICCI和平均曲率流（MCF）的奇异性和刚性。 平均曲率流是一种非线性抛物线进化方程，起源于材料科学，并在纯数学和应用数学中进行了深入研究。封闭的表面会演变成尽可能有效地降低其面积，从而使自己紧绷。随着表面变小，流动收缩甚至更快，因此总是会出现奇点。关键是要了解奇异性。功能理论扮演着连续和离散和独特的延续。第二个主要方向是了解RICCI流动中奇点的某些特性，包括爆炸是独一无二的，爆炸是严格的，以及梯度缩小的渐进式缩小的Ricci Solitons的渐近结构。该奖项反映了NSF的法定任务，并通过该基金会的知识优点和广泛的影响来评估NSF的法定任务，并通过评估值得进行评估。