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Geometric Variational Problems and Nonlinear Partial Differential Equations

几何变分问题和非线性偏微分方程

基本信息

批准号：
1811034
负责人：
Matthew Gursky
金额：
$ 30.23万
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：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2022-09-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811034&HistoricalAwards=false
关键词：
Geometric Variational Problems Nonlinear Partial

项目摘要

The interaction of geometry and analysis date back to at least the eighteenth century, and yet continues to be an important and highly active field of mathematical research. The classical subject of geometry grew out of our desire to understand certain properties of the physical world such as angles, distances and properties of certain shapes. Differential geometry was developed to use the tools of calculus to understand the geometry of curved spaces--for example, the curvature of space by matter as predicted by general relativity, or the properties of soap bubbles (which turn out to be related to the equations describing black holes). In the same way that Descartes realized that plane geometry can be studied using algebra, so differential geometry can be studied using techniques from analysis, especially differential equations. The research in this project involves disparate problems from geometry and mathematical physics but are united by the role played by mathematical analysis in their study. In addition to mathematical investigations, the PI will be organizing a summer residential STEM program in cooperation with the Chicago Pre-College Science and Engineering Program, and supported by the Notre Dame TRiO Program. This will be a two-week program run in the summers of 2019 and 2020 for high school students from 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 supported by this award. Poincare-Einstein manifolds are generalizations of the Poincare 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 supported by this award is the fundamental question of existence: given a manifold with boundary and a conformal class of metrics on the boundary, can one construct a Poincare-Einstein metric whose compactification induces the given conformal class on the boundary? In joint work with Q. Han and S. Stolz, the PI is developing new tools to detect obstructions to existence based on the topology and geometry of the boundary. On the other hand, in work with G. Szekelyhidi the PI was able to prove local existence of solutions (i.e., in a neighborhood of infinity). Another area of research with connections to physics is the PI's ongoing work with J. Streets and C. Kelleher on the variational properties of the Yang-Mills functional in four dimensions. Building on the recent work, which gave a new sharp lower bound for minimizing solutions, the PI will investigate the behavior of solutions with large Morse index. The PI will prove a lower bound for the energy depending (in a precise way) on the index of the solution and the geometry of the base manifold.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.

几何与分析的相互作用至少可以追溯到世纪，至今仍是数学研究中一个重要而活跃的领域。几何学的经典主题源于我们对理解物理世界的某些属性的渴望，例如角度，距离和某些形状的属性。微分几何的发展是为了使用微积分的工具来理解弯曲空间的几何形状-例如，广义相对论预测的物质空间的曲率，或者肥皂泡的性质（结果与描述黑洞的方程有关）。在同样的方式，笛卡尔意识到，平面几何可以研究使用代数，所以微分几何可以研究使用技术分析，特别是微分方程。该项目的研究涉及几何和数学物理的不同问题，但通过数学分析在其研究中所发挥的作用而统一起来。除了数学调查，PI还将与芝加哥大学预科科学与工程项目合作，并在圣母大学TRIO项目的支持下，组织一个夏季住宅STEM项目。这将是一个为期两周的计划，在2019年和2020年夏天运行的高中学生从芝加哥公立学校，其中许多人将是第一代大学生。该计划将持续两周，将包括数学教学和基于项目的学习。该奖项支持两个主要的数学主题。庞加莱-爱因斯坦流形是双曲空间的庞加莱球模型的推广。它们是满足爱因斯坦条件（具有负爱因斯坦常数）的完备流形，可以通过共形地改变在无穷远处以适当速率消失的度量来紧致化。它们出现在数学和理论物理的几个领域;例如，在费曼-格雷厄姆共形不变量理论和量子场论中的AdS/CFT对应中。该奖项支持的一个调查领域是存在的基本问题：给定一个有边界的流形和边界上的共形类度量，可以构造一个庞加莱-爱因斯坦度量，其紧化导致边界上的给定共形类吗？与Q合作。Han和S. PI正在开发新的工具，以根据边界的拓扑结构和几何形状来检测存在的障碍物。另一方面，在与G. Szekelyhidi PI能够证明解的局部存在性（即，在无穷大的邻域中）。另一个与物理学有关的研究领域是PI正在与J. Streets和C. Kelleher关于四维Yang-Mills泛函的变分性质。基于最近的工作，这给了一个新的尖锐的下限最小化的解决方案，PI将调查的行为与大莫尔斯指数的解决方案。 PI将证明一个能量的下限，该下限取决于（以精确的方式）解决方案的指数和基础流形的几何形状。该奖项反映了NSF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。