Hessian equations with geometric applications

本项目继续主要研究者对微分几何和非线性偏微分方程之间相互作用的研究。特别是，它将研究某些类型的Hessian方程的正则性，包括四阶椭圆型方程，以及欧氏空间中体积最小化的拉格朗日子流形，Calabi-Yau流形，以及某些伪黎曼流形。在过去的五年里，关于二阶特殊拉格朗日方程的许多问题，包括存在性和正则性，都取得了很大的进展。在这个项目中，首席研究员打算研究这个方程的四阶推广，它可能与物理更相关。非线性四阶偏微分方程代表着一个年轻而令人兴奋的领域，任何进展都可能适用于其他方程，甚至其他物理科学。最近和正在进行的最优运输理论的发展表明，最优运输地图的结构与伪黎曼空间中某些极大定标子流形的几何有关。该项目的一个目标是应用校准流形的机制(如特殊的拉格朗日)来获得最优传输的新结果，从而形成一幅漂亮的几何图。该项目涉及到两个令人兴奋的数学领域的研究，这两个领域似乎在表面下联系在一起。最优运输问题提出了如何在两个地点之间以最低成本最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。发展强大的数学理论，强调关键要素，使工业中的那些人能够实施基于该理论的解决方案。例如，从事物流工作的人可能想知道运送某些商品最便宜的方式。用暴力解决这个问题在计算上可能是不可能的，但如果有一个好的数学理论，就可以有效地计算出解决方案。优化交通的另一个建议用途是创建实时辅助外科医生的软件。要做到这一点，就需要一个坚实的理论。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，我们应该尝试理解被称为拉格朗日子流形的对象。这些对象就像极小曲面一样，具有特殊的性质，并受非线性方程的支配。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未来的物理学研究产生影响。最近，主要研究人员和他的合作者将寻找最优运输地图的问题与描述某种类型的拉格朗日极小曲面的问题联系起来。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。这个项目现在想把几何学中的一些想法应用到最优运输理论中。

This project continues the principal investigator's study of interactions between differential geometry and nonlinear partial differential equations. In particular, it will investigate regularity for certain classes of Hessian equations, including fourth-order elliptic equations, and also volume-minimizing Lagrangian submanifolds of Euclidean space, Calabi-Yau manifolds, and certain pseudo-Riemannian manifolds. Many questions on second-order special Lagrangian equations, including existence and regularity, have witnessed great progress over the last five years. In this project, the principal investigator intends to study the fourth-order generalization of this equation, which may be even more relevant to physics. Nonlinear fourth-order partial differential equations represent a young and exciting field, and any progress may be adaptable to other equations and even other physical sciences. Recent and ongoing developments in the theory of optimal transportation show that the structure of the optimal transportation map is related to the geometry of certain maximal calibrated submanifolds of a pseudo-Riemannian space. A goal of the project is to apply the machinery of calibrated manifolds (such as special Lagrangian) to obtain novel results in optimal transport, thereby developing a nice geometric picture.This project involves research into two exciting areas of mathematics, which appear to be linked together under the surface. The optimal transport problem asks the question of how to transport materials most cost efficiently between two locations. The answers are directly applicable in many areas of science, including economics, medical imaging, fluid mechanics, and meteorology. Development of a strong mathematical theory that emphasizes the crucial elements allows those in industry to implement solutions based on the theory. For example, someone working in logistics may want to know the cheapest way to ship certain goods. Solving the problem by brute force may not be computationally possible, but if a good mathematical theory is available, the solution can be computed efficiently. Another proposed use of optimal transportation is to create software that assists surgeons in real time. In order for this to happen, a solid theory is necessary. 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 get a better understanding of string theory, we should try to understand objects called Lagrangian submanifolds. These objects are like minimal surfaces that have special properties and are governed by nonlinear equations. 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 these questions will impact the study of physics going forward. Recently, the principal investigator and his collaborators have related the problem of finding the optimal transportation map to the problem of describing a certain type of Lagrangian minimal surface. The smoothness of minimal surfaces has been intensely studied by mathematicians for decades. This project would now like to apply some of the ideas from geometry to the theory of optimal transport.