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.
