Microlocal Analysis and Monge-Ampère Type Equations in Geometry

本研究计画关注微分几何中可以用非线性偏微分方程式来表述的问题。其中一个研究主题是空间或流形上规范几何或形状的存在，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。其中一个例子是存在具有圆锥奇点的凯勒-爱因斯坦度量;这些结构在数学和物理学中至关重要，其理论涉及代数、分析、几何和拓扑学的发展。在这个项目中开发的分析技术预计将是有用的研究人员在几何，物理和相关领域的工作。此外，该项目旨在更好地理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，推广勒让德变换的理论，这是数学，力学和经济学中的经典工具。该项目包括对研究生进行相关主题的研究培训。了解具有圆锥奇点的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出现江口-汉森度量的理论物理。Monge-Ampère型方程出现在纯数学和应用数学中的各种问题中，并具有广泛的实际应用。该项目旨在开发构造和近似这些解的方法，并研究其规律性，这将在这些方程出现的其他情况下得到应用。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的连接。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.