Microlocal Analysis and Monge-Ampere Type Equations in Geometry

几何中的微局域分析和Monge-Ampere型方程

• 批准号: 1906370
• 负责人: Yanir Rubinstein
• 金额: $ 34.25万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1906370&HistoricalAwards=false
• 关键词: Microlocal; Analysis; Monge; Ampere; Type

In this project the PI will continue to study problems mainly in differential geometry that can be formulated as highly nonlinear partial differential equations. One of the themes is the existence of canonical geometries or shapes on spaces. These originally grew out of Einstein's famous equation in general relativity. One example is the existence of Kahler-Einstein metrics with conic singularities. These beautiful structures turn out to be of central importance in mathematics and physics, and touch upon many fields and their theory involves progress relevant to algebraic, analysis, and geometry and topology. The analytic techniques developed in this proposal should be useful to researchers working in geometry, physics and elsewhere. Also, developing a better understanding for the polarity transform in convex geometry could be useful to solving a range of partial differential equations, and generalizes the known theory for the Legendre transform that is a classical tool in mathematics, mechanics and economics. The award also supports graduate students working on their dissertations in related topics.Understanding Kahler-Einstein metrics with conic singularities will deepen our understanding of smooth Kahler-Einstein metrics on both compact and non-compact Kahler manifolds, including Fano and Calabi-Yau spaces. These spaces are central in a wide variety of fields, ranging from algebraic geometry and number theory to theoretical physics. Monge-Ampere type equations arise in a wide variety of problems in pure and applied mathematics and have a wide range of real-world applications. Developing methods and techniques to construct and approximate such solutions and to study their regularity could have applications in other instances where these equations appear. Moreover, developing novel connections with algebraic geometry, convex geometry, and micro-local analysis will be an important goal of this project.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.

在这个项目中，PI将继续主要研究微分几何中的问题，这些问题可以被表述为高度非线性的偏微分方程。其中一个主题是规范几何或空间形状的存在。它们最初是由爱因斯坦在广义相对论中著名的方程推导出来的。一个例子是具有二次奇点的卡勒-爱因斯坦度量的存在性。这些美丽的结构在数学和物理学中至关重要，涉及许多领域，它们的理论涉及与代数、分析、几何和拓扑相关的进展。在这个提议中发展的分析技术应该对研究几何、物理和其他领域的研究人员有用。此外，更好地理解凸几何中的极性变换可能有助于解决一系列偏微分方程，并推广已知的勒让德变换理论，这是数学，力学和经济学中的经典工具。该奖项还支持在相关主题上撰写论文的研究生。理解具有二次奇点的Kahler- einstein度量将加深我们对紧致和非紧致Kahler流形（包括Fano和Calabi-Yau空间）上的光滑Kahler- einstein度量的理解。这些空间是各种领域的中心，从代数几何和数论到理论物理。蒙日-安培型方程出现在纯数学和应用数学的各种各样的问题，并有广泛的现实世界的应用。开发方法和技术来构造和近似这些解，并研究它们的规律性，可以应用于这些方程出现的其他情况。此外，发展与代数几何、凸几何和微局部分析的新联系将是本项目的一个重要目标。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。