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）资助的。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。特别地，它将探索(A)某些非线性Hessian方程的正则性和(B)欧几里得空间的特殊lagrange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低相中不成立。首席研究员袁宇的工作完成了三维图像，但在已知的大于三维的空间中仍然存在空白。PI和他的合作者也在寻找其他相关对称黑森方程的规律性。最近和正在进行的最优运输理论的发展表明，最优运输图的正则性与伪黎曼空间的某些最大校准子流形的几何形状有关。该项目的目标是应用校准歧管的机制来获得最佳传输的新结果，在此过程中形成一个漂亮的几何图像。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。在20世纪90年代末，顶尖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”的物体。这些物体是具有特殊性质的最小表面，由非线性方程控制。这个项目试图回答诸如这些表面何时是光滑的，何时是平坦的，以及何时是不连续的等问题。这些问题的答案将对基础物理学的研究产生影响。最优运输问题的问题是如何在两个地点之间最经济有效地运输材料。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。也许最大的问题是这样的：什么时候最优运输是连续的？最近，首席研究员发现了寻找最优交通地图问题与描述某一类特殊拉格朗日最小曲面问题之间的联系。几十年来，数学家们对最小曲面的光滑性进行了深入的研究。这个项目现在试图将几何学中的一些思想应用到最佳运输理论中。

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.