Microlocal Analysis and Monge-Ampere Type Equations in Geometry

在这个项目中，PI将继续主要研究可以表示为高度非线性偏微分方程组的微分几何问题。其中一个主题是空间中规范几何或形状的存在。这些起源于爱因斯坦在广义相对论中著名的方程式。一个例子是具有圆锥奇点的卡勒-爱因斯坦度规的存在。这些美丽的结构在数学和物理中具有中心重要性，涉及许多领域，它们的理论涉及与代数、分析、几何和拓扑学相关的进展。这项建议中开发的分析技术应该对从事几何、物理和其他领域工作的研究人员有用。此外，更好地理解凸几何中的极性变换有助于求解一系列偏微分方程，并推广了勒让德变换的已知理论，勒让德变换是数学、力学和经济学中的经典工具。该奖项还支持研究生从事相关主题的论文工作。理解具有二次奇点的Kahler-Einstein度量将加深我们对紧致和非紧Kahler流形上的光滑Kahler-Einstein度量的理解，包括Fano和Calabi-Yau空间。这些空间在从代数几何和数论到理论物理的广泛领域中都是中心。Monge-Ampere型方程出现在纯数学和应用数学中的各种问题中，并在现实世界中有广泛的应用。开发方法和技术来构造和近似这种解并研究它们的正则性，可以在出现这些方程的其他情况下应用。此外，发展与代数几何、凸几何和微观局部分析的新联系将是该项目的重要目标。该奖项反映了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。

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.