Microlocal Analysis and Monge-Ampere Type Equations in Geometry

这个项目主要集中在几何分析中的问题，这些问题可以被表述为广义的Monge-Ampere类型的偏微分方程。总的来说，本提案中开发的分析和几何技术对几何、物理和其他领域的研究人员应该很有用。一方面，加深我们对卡勒流形上的正则几何和卡-丘流形上的拉格朗日量的理解，似乎是试图模拟宇宙几何的物理学家感兴趣的。另一方面，这些规范几何与数学中各种各样的既定领域有关系。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，并且具有广泛的现实应用，例如气象学和网络的优化设计。开发方法和技术来构造和近似解这些方程，并研究其规律性，可以在这些方程出现的其他情况下应用。最后，勒让德变换是数学、力学和经济学中的经典工具，寻求将这一理论推广到其他环境，如本项目中，可以找到广泛的应用。本项目中提出的一些方程是新的，不属于传统的Monge-Ampere型方程，并且具有令人兴奋的新几何应用，需要新的分析工具。在这项研究中的一个新的特点是应用几种不同的微局部分析工具，传统上属于线性问题，研究这些完全非线性方程。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。研究的问题包括：（1）具有圆锥奇点的Kahler-Einstein度规。这些度量为代数几何和复几何提供了一种新的强有力的分析工具。(2)PI将研究特殊拉格朗日方程的新退化版本。它支配着卡-丘流形上的正拉格朗日空间中的测地线。了解这个方程的解决方案需要新的方法，并将有特殊的拉格朗日的存在性和唯一性，奇异的拉格朗日平均曲率流，拓扑空间的拉格朗日，和拉格朗日相交理论的应用。(3)PI将使用空间中的测地线，有限维Bergman近似，具有复相位的傅立叶积分算子以及该空间的度量空间几何来研究Kahler度量的空间。(4)PI正在开发复杂和凸几何之间的相互作用，包括与勒让德变换的已知理论并行的极性变换的微分理论。由此产生的新的方程的哈密顿-雅可比和蒙日-安培型将被调查，以及复杂的类似物。这些方程提供了新的Banach空间之间的插值过程和新的概念的最佳运输。

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.