Microlocal Analysis and Monge-Ampère Type Equations in Geometry

这个研究项目涉及可以用非线性偏微分方程组表示的微分几何中的问题。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，这与黎曼关于曲率和爱因斯坦广义相对论方程的原始工作有关。一个例子是具有圆锥奇点的Kähler-Einstein度量的存在；这些结构被证明在数学和物理中具有核心重要性，它们的理论涉及代数、分析、几何和拓扑学的发展。这个项目中将要开发的分析技术有望对从事几何、物理和相关领域工作的研究人员有用。此外，该项目旨在更好地了解复杂的勒让德变换，这可能有助于解决一系列偏微分方程式，推广勒让德变换的理论，勒让德变换是数学、力学和经济学中的经典工具。该项目涉及研究生相关主题的研究培训。理解具有二次奇点的Kähler-Einstein度量将加深对紧致和非紧Kähler流形(包括Fano和Calabi-Yau空间)上光滑Kähler-Einstein度量的理解。从代数几何和数论到出现江口-汉森度规的理论物理，这些空间在各种各样的领域中都是中心。Monge-Ampère方程出现在纯数学和应用数学中的许多问题中，并有着广泛的实际应用。该项目旨在开发构造和近似此类解的方法，并研究其正则性，这将在出现这些方程的其他情况下应用。该项目还打算发展与代数几何、凸几何和微观局部分析的新联系。这一奖项反映了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。

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.