Microlocal Analysis and Monge-Ampère Type Equations in Geometry

这个研究项目涉及微分几何中可以用非线性偏微分方程表述的问题。其中一个研究主题是空间或流形上的正则几何或形状的存在，与黎曼关于曲率的原始工作和爱因斯坦广义相对论方程有关。一个例子是具有二次奇异性的Kähler-Einstein度量的存在性；这些结构在数学和物理学中是非常重要的，它们的理论涉及代数、分析、几何和拓扑学的发展。在这个项目中开发的分析技术有望对几何、物理和相关领域的研究人员有用。此外，该项目旨在更好地理解复杂的Legendre变换，这可能有助于解决一系列偏微分方程，推广Legendre变换理论，这是数学，力学和经济学中的经典工具。本项目涉及对研究生进行相关课题的研究训练。理解具有二次奇点的Kähler-Einstein度量将加深对紧致和非紧致Kähler流形（包括Fano和Calabi-Yau空间）上光滑Kähler-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Eguchi-Hanson度量出现的理论物理。在纯数学和应用数学中，蒙日-安普雷方程出现在各种各样的问题中，具有广泛的实际应用。本项目旨在开发构建和近似这些解的方法，并研究它们的规律性，这将在这些方程出现的其他实例中得到应用。该项目还打算发展与代数几何、凸几何和微局部分析的新联系。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。

This research project concerns questions in differential geometry that can be formulated in terms of nonlinear partial differential equations. One of the research themes is the existence of canonical geometries or shapes on spaces or manifolds, related to the original work of Riemann on curvature and Einstein's equations of general relativity. One example is the existence of Kähler-Einstein metrics with conic singularities; these structures turn out to be of central importance in mathematics and physics, and their theory involves developments in algebra, analysis, geometry, and topology. The analytic techniques to be developed in this project are expected to be useful to researchers working in geometry, physics, and related areas. Additionally, the project aims to develop better understanding of the complex Legendre transform, which could be useful in solving a range of partial differential equations, generalizing the theory for the Legendre transform that is a classical tool in mathematics, mechanics, and economics. The project involves research training of graduate students in related topics.Understanding Kähler-Einstein metrics with conic singularities will deepen understanding of smooth Kähler-Einstein metrics on both compact and non-compact Kähler 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 where the Eguchi-Hanson metric appears. Monge-Ampère type equations arise in a wide variety of questions in pure and applied mathematics and have a wide range of practical applications. This project aims to develop methods to construct and approximate such solutions and to study their regularity, which will have applications in other instances where these equations appear. The project also intends to develop novel connections with algebraic geometry, convex geometry, and micro-local 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.