Microlocal Analysis and Monge-Ampere Type Equations in Geometry

这个项目主要关注几何分析中的问题，这些问题可以用广义上理解的Monge-Ampere型偏微分方程组来表示。总的来说，这项建议中发展的解析和几何技术对从事几何、物理和其他领域工作的研究人员应该是有用的。一方面，加深我们对Kahler流形上的正则几何和Calabi-Yau流形中的拉格朗日几何的理解，似乎是试图对宇宙几何建模的物理学家感兴趣的。另一方面，这些正则几何与数学中的许多既定领域都有关系。此外，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，如气象学和网络优化设计。发展方法和技术来构造和逼近这些方程的解，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。最后，勒让德变换是数学、力学和经济学中的经典工具，寻求将该理论推广到其他环境中，如在本项目中，可以找到广泛的应用。本项目中提出的一些方程是新的，属于传统已知的Monge-Ampere型方程，并且具有激动人心的新几何应用，需要新的分析工具。这项研究的一个新特点是应用几种不同的微局部分析工具，传统上与线性问题有关，来研究这些完全非线性的方程。另一个主题是研究凸分析和几何以及复分析和几何之间的新关系。所研究的问题包括：(1)具有二次曲线奇异性的Kahler-Einstein度规。这些度量为代数和复杂几何提供了一种新的强大的分析工具。(2)PI将研究特殊拉格朗日方程的一个新的简并版本。它控制Calabi-Yau流形上正拉格朗日空间中的测地线。理解这个方程的解需要新的方法，并将在特殊拉格朗日的存在唯一性、拉格朗日平均曲率流的奇性、拉格朗日空间的拓扑以及拉格朗日交集理论中得到应用。(3)PI应研究空间中使用测地线的Kahler度量空间、有限维Bergman逼近、具有复位相的傅立叶积分算子以及该空间的度量空间几何。(4)PI正在发展复杂几何和凸几何之间的相互作用，包括与勒让德变换的已知理论平行的极性变换的微分理论。由此产生的新的哈密顿-雅可比方程和蒙格-安培型方程将被研究，以及复杂的类似物。这些方程为Banach空间之间的内插和最优传输的新概念提供了新的处理方法。

This project focuses on problems mainly in geometric analysis that can be formulated as partial differential equations of Monge-Ampere type, broadly understood. In general terms, the analytic and geometric techniques developed in this proposal should be useful to researchers working in geometry, physics, and elsewhere. On the one hand, deepening our understanding of canonical geometries on Kahler manifolds and Lagrangians in Calabi-Yau manifolds seems to be of interest to physicists trying to model the geometry of the universe. On the other hand, these canonical geometries have relations to a wide variety of established fields in mathematics. Moreover, 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, such as meteorology and optimal design of networks. Developing methods and techniques to construct and approximate solutions to such equations and to study their regularity could have applications in other instances where these equations appear. Finally, the Legendre transform is a classical tool in mathematics, mechanics, and economics, and seeking generalizations of this theory to other settings, as in this project, could find a broad range of applications.A number of the equations proposed in this project are new, fall outside of the traditionally known Monge-Ampere type equations, and have exciting new geometric applications requiring new analytical tools. A novel feature in this research is to apply several different tools of microlocal analysis, traditionally pertaining to linear problems, to study these fully nonlinear equations. Another theme is to investigate novel relations between convex analysis and geometry and complex analysis and geometry. The problems investigated include: (1) Kahler-Einstein metrics with conic singularities. These metrics provide a new powerful analytic tool in algebraic and complex geometry. (2) The PI will study a new degenerate version of the special Lagrangian equation. It governs geodesics in the space of positive Lagrangians on a Calabi-Yau manifold. Understanding solutions of this equation requires new methods and will have applications to existence and uniqueness of special Lagrangians, singularities of Lagrangian mean curvature flow, the topology of the space of Lagrangians, and Lagrangian intersection theory. (3) The PI shall study the space of Kahler metrics using geodesics in the space, finite-dimensional Bergman approximations, Fourier integral operators with complex phase, and the metric space geometry of this space. (4) The PI is developing interactions between complex and convex geometry, including a differential theory for the polarity transform in parallel to the known theory for the Legendre transform. New equations of Hamilton-Jacobi and Monge-Ampere type that arise from this will be investigated as well as complex analogues. These equations provide new processes for interpolation between Banach spaces and new notions of optimal transportation.