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.

该职业奖将支持一项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。研究目标围绕对曲率不均匀的形状有更深入的了解。平面中两个点之间的距离由段的长度连接。该部分可以用尺子构建，但是在任意几何形状上找到两个点之间的距离要困难得多，部分原因是这种形状不伴随着标尺！找到这种“统治者”，即测量几何形状距离的最佳方法，植根于数学物理学的深层问题。在数学中，我们使用指标测量距离。当试图找到理想的指标时，通常必须找到一个解决特定偏微分方程的平滑函数。这是一个优化问题，其最小化器正是部分微分方程的解决方案。可以将非平滑函数插入动作功能（称为电势）中，为通常称为基础方程的潜在理论打开了大门。该项目涉及复杂几何形状中的问题，在复杂的几何形状中，可以给予所考虑的电位非常特定的度量几何形状，从而使人们比平时更加​​微妙的理解。除了拟议的研究外，该项目还将遵循垂直整合的教育计划，其中包括各种形式的公共外展普及的STEM领域（例如创建教育视频并在线发布），进行本科夏季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。 第一部分专门介绍了Kahler指标空间内的地球射线的几何潜在理论，以期在复杂几何形状中存在规范的Kahler指标的各种特征。通过足够开发的大地测量光线的度量几何形状，人们可以将这些猜想视为在大地射线空间上的优化问题，并考虑了有关射线的规律性的各种改进。第二部分专门介绍了奇点类型空间的几何潜在理论，并将其视为具有规定的奇异性和乘数理想带的奇异性和变化的复杂蒙格 - 安培方程。预计奇异性类型的度量空间将是完整的，它将允许研究奇异性的奇异Kahler-Einstein指标。在最后一部分中，我们研究了Kahler几何学研究与几何分析的其他部分的研究相互作用，包括Hermitian几何学和凸分析。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响审查标准通过评估来进行评估的。