Geometric Variational Problems and Nonlinear Partial Differential Equations

几何和分析的相互作用至少可以追溯到18世纪，但仍然是数学研究的一个重要和高度活跃的领域。几何这门经典学科源于我们想要理解物理世界的某些特性，比如角度、距离和某些形状的特性。微分几何的发展是为了利用微积分的工具来理解弯曲空间的几何——例如，广义相对论所预测的物质对空间的曲率，或者肥皂泡的性质（结果与描述黑洞的方程有关）。就像笛卡尔意识到平面几何可以用代数来研究一样，微分几何也可以用分析的方法来研究，尤其是微分方程。该项目的研究涉及几何和数学物理的不同问题，但数学分析在它们的研究中所起的作用将它们联系在一起。除了数学研究之外，PI还将与芝加哥大学预科科学与工程项目合作组织一个夏季住宿STEM项目，并得到圣母大学三重奏项目的支持。这将是一个为期两周的项目，将于2019年和2020年夏季为芝加哥公立学校的高中生开设，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。庞加莱-爱因斯坦流形是双曲空间庞加莱球模型的推广。它们是满足爱因斯坦条件（负爱因斯坦常数）的完全流形，可以通过在无穷远处以适当的速率消失的度规的共形改变而紧化。它们出现在数学和理论物理的几个领域；例如，在费弗曼-格雷厄姆共形不变量理论和量子场论的AdS/CFT对应中。该奖项支持的一个研究领域是存在的基本问题：给定一个具有边界的流形和边界上的共形度量类，是否可以构造一个庞加莱-爱因斯坦度量，其紧化在边界上诱导出给定的共形类？在与Q. Han和S. Stolz的合作中，PI正在开发基于边界拓扑和几何形状的新工具来检测存在的障碍物。另一方面，在与G. Szekelyhidi的合作中，PI能够证明解的局部存在性（即在无穷邻域内）。另一个与物理学相关的研究领域是PI与J. Streets和C. Kelleher正在进行的关于四维杨-米尔斯泛函的变分性质的研究。在最近的工作的基础上，给出了最小化解的一个新的尖锐下界，PI将研究具有大莫尔斯指数的解的行为。PI将证明能量的下界（以一种精确的方式）取决于解的指数和基流形的几何形状。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。

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.