CAREER: Geometric Potential Theory

职业：几何势理论

• 批准号: 1846942
• 负责人: Tamas Darvas
• 金额: $ 43.05万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1846942&HistoricalAwards=false
• 关键词: CAREER; Geometric; Potential; Theory

This CAREER award will support a multifaceted program of research and education aiming at significant progress in both areas. The research goals center around a deeper understanding of shapes with uneven curvature. The distance between two points in the plane is given by the length of the segment joining them. The segment can be constructed with a ruler, however finding the distance between two points on an arbitrary geometric shape is much more difficult, partly because such shapes don't come with a ruler! Finding this "ruler", i.e., the best possible way to measure distances on geometric shapes, is rooted in deep problems of mathematical physics. In mathematics we measure distances using metrics. When trying to find ideal metrics, one often has to find a smooth function that solves a specific partial differential equation. This is an optimization problem with an action functional whose minimizers are exactly the solutions of the partial differential equation. It is possible to plug in non-smooth functions into the action functional, called potentials, opening the door to what is often referred to as the potential theory of the underlying equation. This project deals with problems in complex geometry where the potentials considered can be given a very specific metric geometry, leading to a much more delicate understanding than usual. In addition to the proposed research, the project will pursue a vertically integrated educational program that includes various forms of public outreach popularizing STEM fields (such as creating educational videos and posting them online), conducting undergraduate summer research and the involvement of graduate students.The research goals of the project can be split in three parts. The first part is devoted to the geometric potential theory of the geodesic rays inside the space of Kahler metrics, with a view toward various characterizations for existence of canonical Kahler metrics in complex geometry. With the metric geometry of geodesic rays sufficiently developed, one can look at these conjectures as optimization problems on the space of geodesic rays, with various refinements on the regularity of the rays considered. The second part is devoted to the geometric potential theory of the space of singularity types, with a view toward complex Monge-Ampere equations with prescribed singularity and variation of multiplier ideal sheaves. The metric space of singularity types is expected to be complete, and it will allow for a study of singular Kahler-Einstein metrics, under variation of the singularity. In the last part we study interactions of the investigations in Kahler geometry with other parts of geometric analysis, including Hermitian geometry and convex analysis.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.

该职业奖将支持多方面的研究和教育计划，旨在在这两个领域取得重大进展。研究目标围绕更深入地了解曲率不均匀的形状。平面中两点之间的距离由连接它们的线段的长度给出。该线段可以用尺子构建，但是找到任意几何形状上两点之间的距离要困难得多，部分原因是此类形状没有尺子！找到这个“标尺”，即测量几何形状上的距离的最佳方法，植根于数学物理的深层问题。在数学中，我们使用度量来测量距离。当试图找到理想的度量时，人们通常必须找到一个求解特定偏微分方程的平滑函数。这是一个具有作用函数的优化问题，其极小值恰好是偏微分方程的解。可以将非光滑函数插入称为势的动作函数中，从而为通常所说的基础方程的势理论打开了大门。该项目处理复杂几何中的问题，其中所考虑的潜力可以被赋予非常具体的度量几何，从而导致比平常更微妙的理解。除了拟议的研究之外，该项目还将推行垂直整合的教育计划，包括各种形式的公共宣传推广 STEM 领域（例如制作教育视频并将其发布到网上）、开展本科生暑期研究以及研究生的参与。该项目的研究目标可以分为三个部分。 第一部分致力于卡勒度量空间内测地射线的几何势理论，着眼于复杂几何中规范卡勒度量存在的各种表征。随着测地射线的度量几何的充分发展，人们可以将这些猜想视为测地射线空间上的优化问题，并考虑对射线规律性的各种细化。第二部分致力于奇点类型空间的几何势理论，着眼于具有指定奇点和乘法器理想滑轮变化的复杂蒙日-安培方程。奇点类型的度量空间预计是完整的，并且它将允许在奇点变化的情况下研究奇异卡勒-爱因斯坦度量。在最后一部分中，我们研究卡勒几何的研究与几何分析的其他部分（包括埃尔米特几何和凸分析）之间的相互作用。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响进行评估，被认为值得支持审查标准。