Calculus of variations and optimal mass transportation: theory and applications

这项建议主要集中在两个主要方向，偏微分方程组，重点是变分演算，以及最优质量传输的理论和应用。偏微分方程(PDE)因其与物理世界中现象的联系而引起人们的广泛关注。科学家和数学家积极参与研究由偏微分方程组模拟的无数问题。我主要对自然科学和微分几何中产生的非线性偏微分方程组的研究感兴趣。除了开发新的技术外，我还应用泛函和凸分析的工具来研究确定性和随机偏微分方程组的定性性质。并对最优公共交通的理论和应用进行了研究。最优质量传输是一个历史悠久的学科，有着令人难以置信的丰富的理论和应用，它定义了概率分布之间的稳健度量。通过相关的运输计划计算分布之间的最佳位移，使这一理论成为包括医学图像处理、癌症检测和统计机器学习在内的几个适用领域的主流。这是一个充满活力和不断增长的领域，导致了令人惊讶的新发现以及对经典结果的新颖解释。这项拟议的研究计划的目的有两个。首先，将变分和变分不等式的技巧与新建立的变分原理相结合，研究和分析了自然科学中产生的一大类随机和确定性偏微分方程解的存在性和定性性质。第二，刻画和仔细研究标准和鞅多边际最优质量运输问题的解，因为它们有能力为信号强度和其他数据分布提供准确的模型。除了发展最优运输的数学理论外，我还将致力于它的各种现实应用。我期待这项研究的成果将在偏微分方程组、几何测度论、变分法、随机偏微分方程组以及最优公共交通在其他领域的应用方面产生重大影响。

This proposal is focused on two primary directions, partial differential equations with an emphasis on the calculus of variations, and the theory and applications of optimal mass transportation. Partial differential equations (PDEs) are of widespread interest because of their connection with phenomena in the physical world. Scientists and mathematicians have become actively involved in the study of countless problems modeled by PDEs.  I am mainly interested in the study of non-linear partial differential equations arising from natural sciences and differential geometry. In addition to developing new techniques, I apply tools from functional and convex analysis to study the qualitative properties of deterministic and stochastic partial differential equations. I also study the theory and application of optimal mass transportation. Optimal mass transport is a subject with a long history, with incredibly rich theory and applications, that defines robust metrics between probability distributions. The computation of optimal displacements between distributions through an associated transport plan makes this theory mainstream in several applicable fields including image processing, cancer detection, and statistical machine learning. It is a dynamic and growing area that has led to surprising new discoveries as well as novel interpretations of classical results. The purpose of this proposed research program is two-fold. First, to combine techniques from the calculus of variations and variational inequalities with newly established variational principles to study and analyze the existence, and qualitative properties of solutions for a large class of stochastic and deterministic partial differential equations arising from the natural sciences. Second, to characterize and scrutinize solutions of standard and martingale multi-marginal optimal mass transport problems given their abilities to provide accurate models for signal intensities and other data distributions. In addition to developing the mathematical theory of optimal transportation, I will also work on its various real-life applications. I expect the outcomes of this research to have significant impacts in the fields of partial differential equations, geometric measure theory, calculus of variations, stochastic PDEs, and applications of optimal mass transportation in other fields.