Hessian and Special Lagrangian Equations

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)提供资金的。该项目将研究微分几何和非线性偏微分方程之间的相互作用。特别是，它将探索(A)某些类型的非线性Hessian方程的正则性和(B)欧氏空间的特殊拉格朗日子流形、Calabi-Yau流形和某些伪黎曼流形的几何。最近的一个反例表明，特殊拉格朗日方程的正则性在较低的阶段不成立。首席调查员袁宇的工作完成了这幅三维图像，但在三维以上的已知维度上仍有差距。PI和他的合作者还在寻找其他相关对称Hessian方程的正则性。最优运输理论的最新发展表明，最优运输映射的正则性与伪黎曼空间中某些极大定标子流形的几何有关。该项目的目标是应用校准流形的机械来获得最优传输的新结果，在这个过程中绘制出一幅漂亮的几何图像。弦理论是物理学中一个令人兴奋的发展分支，许多人希望它将导致对宇宙基本相互作用的理解。20世纪90年代末，领先的数学物理学家断言，为了更好地理解弦理论，人们应该首先尝试理解被称为“特殊拉格朗日子流形”的对象。这些对象是具有特殊属性且受非线性方程控制的极小曲面。这个项目试图回答诸如这些表面何时光滑、何时平坦以及何时不连续等问题。这些问题的答案将对基础物理学的研究产生影响。最优运输问题提出了如何在两个地点之间以最低成本有效地运输物资的问题。这些答案直接适用于许多科学领域，包括经济学、医学成像、流体力学和气象学。也许最大的问题是：最佳运输是什么时候是连续的？最近，主要研究者发现了寻找最优运输地图的问题与描述某种特殊拉格朗日极小曲面的问题之间的联系。几十年来，极小曲面的光滑性一直是数学家们深入研究的问题。这个项目现在试图将几何学中的一些想法应用到最优运输理论中。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The project will investigate interactions between differential geometry and nonlinear partial differential equations. In particular, it will explore (A) regularity for certain classes of nonlinear Hessian equations and (B) geometry of special Lagrangian submanifolds of Euclidean space, Calabi-Yau manifolds, and certain pseudo-Riemannian manifolds. A recent counterexample shows that regularity for the special Lagrangian equation does not hold in lower phases. Work of the principal investigator with Yuan Yu then completes the picture in three dimensions, but there is still a gap in what is known for dimensions larger than three. The PI and his collaborators are also looking for regularity of other related symmetric Hessian equations. Recent and ongoing developments in the theory of optimal transportation demonstrate that regularity of the optimal transportation map is related to the geometry of certain maximal calibrated submanifolds of a pseudo-Riemannian space. The project's goal is to apply the machinery of calibrated manifolds to obtain novel results in optimal transport, in the process developing a nice geometric picture.String theory is an exciting developing branch of physics, which many hope will lead to an understanding of the fundamental interactions of the universe. In the late 1990s, leading mathematical physicists asserted that, in order to obtain a better understanding of string theory, one should first try to understand objects called "special Lagrangian submanifolds." These objects are minimal surfaces that have special properties and are governed by a nonlinear equation. This project attempts to answer questions such as when these surfaces are smooth, when they are flat, and when they are discontinuous. The answers to such questions will have an impact on the study of the underlying physics. The optimal transport problem asks the question of how to transport materials most cost effectively between two locations. The answers are directly applicable in many areas of science, including economics, medical imaging, fluid mechanics, and meteorology. Perhaps the biggest question asks the following: When is the optimal transportation continuous? Recently, the principal investigator has found a connection between the problem of finding the optimal transportation map and the problem of describing a certain type of special Lagrangian minimal surface. The smoothness of minimal surfaces has been intensely studied by mathematicians for decades. This project now seeks to apply some of the ideas from geometry to the theory of optimal transport.