Geometric Variational Problems and Nonlinear Partial Differential Equations

几何与分析的相互作用一直是数学研究的一个重要而活跃的领域。几何这门经典学科源于我们想要了解物理世界的属性，如角度和距离。微分几何的发展反过来又是为了利用微积分的工具来理解弯曲空间。例如，正如广义相对论所预言的那样，微分几何可以用来理解物质对空间的曲率。就像平面几何可以用代数来研究一样，微分几何也可以用分析中的技巧来研究。该项目将研究几何和数学物理中的问题，这些问题因数学分析所起的作用而结合在一起。除了数学调查，PI还将继续与芝加哥大学预科科学与工程项目合作，组织夏季住宿STEM项目。这是一个面向芝加哥公立学校高中生的项目，他们中的许多人将是第一代大学生。该计划将运行两周，将包括数学教学和基于项目的学习这个项目有两个主要的数学主题。Poincaré-Einstein流形是双曲空间Poincaréball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，可以通过共形改变在无穷远处以适当速率消失的度规来压缩。它们出现在数学和理论物理的几个领域；例如，在共形不变量的Fefferman-Graham理论中，以及在量子场论中的ADS/CFT对应中。这个项目的一个研究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度量类，人们能否构造一个Poincaré-Einstein度量，它的紧化导致边界上给定的共形类？在与SY.A Chang的合作中，Pi将研究使用共形边界的几何来解决内部的非线性偏微分方程组的问题，并展示解的存在如何允许我们使用Morse理论来识别存在的拓扑障碍。另一个与物理学有关的研究领域是PI与S.Perez-Ayala正在进行的关于共形协变算子的特征值极值的工作。对于曲面，Laplace-Beltrami算子的极值本征值与极小曲面和调和映射有联系。最近与Perez-Ayala Pi的工作表明，在某些自然条件下，人们可以在更高维上极值共形拉普拉斯算子的低本征值，并且存在产生调和映射的极值的例子。在正在进行的工作中，PI将研究其他操作员，并试图理解一种反向构造；即，当调和地图产生最大度量时。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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 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.