Geometric Variational Problems and Nonlinear Partial Differential Equations

几何变分问题和非线性偏微分方程

• 批准号: 2105460
• 负责人: Matthew Gursky
• 金额: $ 28.43万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2105460&HistoricalAwards=false
• 关键词: Geometric; Variational; Problems; Nonlinear; Partial

The interaction of geometry and analysis continues to be an important and active field of mathematical research. The classical subject of geometry grew out of our desire to understand properties of the physical world such as angles and distances. Differential geometry in turn was developed to use the tools of calculus to understand curved spaces. For example, differential geometry can be used to understand the curvature of space by matter as predicted by general relativity. In the same way that plane geometry can be studied using algebra, so differential geometry can be studied using techniques from analysis. The project will investigate problems from geometry and mathematical physics that are united by the role played by mathematical analysis. In addition to mathematical investigations, the PI will continue to organize a summer residential STEM program in cooperation with the Chicago Pre-College Science and Engineering Program. This is a program for high school students from the Chicago Public Schools, many of whom will be first-generation college students. The program will run for two weeks, and will include mathematics instruction and project-based learning There are two main mathematical themes in this project. Poincaré-Einstein manifolds are generalizations of the Poincaré ball model of hyperbolic space. They are complete manifolds satisfying the Einstein condition (with negative Einstein constant) which can be compactified by conformally changing the metric that vanishes at an appropriate rate at infinity. They arise in several areas of mathematics and theoretical physics; for example, in in the Fefferman-Graham theory of conformal invariants and in the AdS/CFT correspondence in quantum field theory. One area of investigation in this project is the fundamental question of existence: given a manifold with boundary and a conformal class of metrics on the boundary, can one construct a Poincaré-Einstein metric whose compactification induces the given conformal class on the boundary? In joint work with S.Y.A Chang, PI will be studying the problem of using the geometry of the conformal boundary to solve a nonlinear PDE in the interior, and showing how the existence of solutions allows us to use Morse Theory to identify topological obstructions to existence. Another area of research with connections to physics is the PI’s ongoing work with S. Perez-Ayala on extremizing eigenvalues of conformally covariant operators. In the case of surfaces, extremal eigenvalues of the Laplace-Beltrami operator have connections to minimal surfaces and harmonic maps. In recent work with Perez-Ayala PI showed that under certain natural conditions, one can extremize the low eigenvalues of the conformal Laplacian in higher dimensions, and there are examples of extremals that give rise to harmonic maps. In ongoing work PI will study other operators and try to understand a kind of reverse construction; i.e., when a harmonic map gives rise to a maximal metric.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.

几何和分析的相互作用仍然是数学研究的重要而活跃的领域。几何学的经典主题源于我们了解物理世界的特性，例如角度和距离。依次开发了差异几何形状来使用计算工具来了解空间。例如，差异几何形状可用于根据一般相对性预测的物质来理解空间的曲率。就像可以使用代数的平面几何形状研究平面几何形状一样，可以使用分析中的技术进行差异几何形状。该项目将调查由数学分析所起的作用团结起来的几何和数学物理学的问题。除了数学投资外，PI还将继续与芝加哥前大学科学和工程计划合作组织夏季居民STEM计划。这是一个针对来自芝加哥公立学校的高中生的计划，其中许多人将是第一代大学生。该计划将运行两个星期，并将包括数学指导和基于项目的学习，该项目中有两个主要的数学主题。 Poincaré-Einstein歧管是双曲线空间的Poincaré球模型的概括。它们是满足爱因斯坦条件的完全歧管（以爱因斯坦常数为负），可以通过共同改变以适当的无限速度消失的度量来压实。它们位于数学和理论物理学的几个领域。例如，在Fefferman-Graham的共形不变性理论中以及量子场理论中的ADS/CFT对应关系中。该项目中的一个调查领域是存在的基本问题：鉴于边界上有一个歧管和边界上的共形指标类别，一个人可以构建一个庞加莱 - 因斯坦公制，其紧凑型诱导边界上给定的保形类别吗？在与S.Y.A Chang的联合合作中，PI将研究使用保形边界几何形状来解决内部非线性PDE的问题，并展示解决方案的存在如何使我们使用Morse理论来识别拓扑对象的存在。与物理学联系的另一个研究领域是PI与S. Perez-Ayala的持续工作，在共同协方差的运营商的极端特征值上。对于表面，Laplace-Beltrami操作员的极端特征值与最小表面和谐波图具有连接。在最近与Perez-ayala Pi的工作中，在某些自然条件下，可以在更高维度的保形拉普拉斯式的低特征值中极端，并且有一些极端的例子引起了谐波图。在正在进行的工作中，PI将研究其他运营商，并尝试了解一种反向结构。即，当谐波地图产生最大度量时，该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响评估标准来评估，被视为珍贵的支持。