Calculus of variations and optimal mass transportation: theory and applications

这个建议是集中在两个主要方向，偏微分方程，重点是变分法，以及最佳质量运输的理论和应用。偏微分方程（PDE）由于其与物理世界中的现象的联系而受到广泛的关注。科学家和数学家们已经积极地参与了对无数由偏微分方程建模的问题的研究。 我主要对自然科学和微分几何中的非线性偏微分方程的研究感兴趣。除了开发新的技术，我应用工具从功能和凸分析研究定性性质的确定性和随机偏微分方程。我还研究了最优质量运输的理论和应用。最优质量传输是一个历史悠久的主题，具有令人难以置信的丰富理论和应用，它定义了概率分布之间的鲁棒度量。通过相关的运输计划计算分布之间的最佳位移，使该理论成为几个应用领域的主流，包括图像处理，癌症检测和统计机器学习。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。这项研究计划的目的是双重的。第一，将变分法和变分不等式的联合收割机技术与新建立的变分原理相结合，研究和分析自然科学中产生的一类随机和确定性偏微分方程解的存在性和定性性质。第二，描述和审查解决方案的标准和鞅多边际最优质量运输问题，因为它们的能力，提供准确的模型信号强度和其他数据分布。除了发展最佳运输的数学理论，我还将致力于其各种现实生活中的应用。我希望这项研究的成果在偏微分方程，几何测度理论，变分法，随机偏微分方程和最优质量运输在其他领域的应用领域产生重大影响。

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.