CAREER: Action-Minimizing Paths in the Space of Probability Measures

这个项目涉及对模拟流体流动的方程系统的数学基础的研究。这些方程的应用包括血液中等离子体的运动，石油在管道中的移动，以及木星和土星等气态行星的形成。这项研究的目的是发展一种方法，可以用来证明在各种物理现象中断言质量和动量守恒的非线性偏微分方程组的解的整体存在性。除了让学生参与这项研究外，这位研究人员还将试行一个“通往博士学位的桥梁”项目，旨在增加在宾夕法尼亚大学获得数学博士学位的女性和少数民族人数。这位调查员还将与费城工作坊学校的教育工作者合作，将数学建模引入他们的课程，并训练学生团队参加由工业和应用数学协会组织的一年一度的穆迪超级数学挑战赛。提出的研究重点是基于广义Euler-Poisson方程，这是一个偏微分方程组，用于研究模拟为连续质量的材料的行为。这些方程在概率度量空间中作为动作的临界点出现；本项目将研究该动作的极小化存在的充要条件。此外，研究人员的目的是为概率度量空间中的拉格朗日力学、变分和控制理论奠定基础。为此，他将使用流体力学、最优输运、泛函不等式和演化方程的方法。调查人员将与宾夕法尼亚大学教学中心合作，组织一次基于这些主题的互动研讨会，作为招募和培训本科生进行研究的一种方式。该项目中提出的教育和外展努力将使研究人员能够激励来自不同背景的各级学生使用数学来应对我们面临的一些最紧迫的科学挑战。

This project involves the study of mathematics underlying systems of equations that model fluid flow. Applications of such equations include the motion of plasma in blood flow, oil moving through pipelines, and the formation of gaseous planets such as Jupiter and Saturn. The aim of the research is to develop methods that can be used to establish the global existence of solutions to systems of nonlinear partial differential equations that assert the conservation of mass and momentum in a variety of physical phenomena. In addition to involving students in the research, the investigator will pilot a "bridge to PhD" program intended to increase the number of women and minorities earning PhD degrees in mathematics at the University of Pennsylvania. The investigator will also work with educators at Philadelphia's Workshop School to introduce mathematical modeling into their curriculum and to train teams of students to compete in the annual Moody's Mega Math Challenge contest for high school juniors and seniors organized by the Society for Industrial and Applied Mathematics. The focus of the proposed research is based on the generalized Euler-Poisson equations, a system of partial differential equations used to study the behavior of materials modeled as continuous masses. These equations arise as critical points of an action in the space of probability measures; the project will investigate sufficient and necessary conditions for minimizers of this action to exist. Moreover, the investigator aims is to lay a foundation for a theory of Lagrangian mechanics, calculus of variations, and control theory in the space of probability measures. To this end, he will employ methods from fluid mechanics, optimal transport, functional inequalities, and evolution equations. In collaboration with the University of Pennsylvania's Center for Teaching and Learning, the investigator will organize an interactive seminar based on these topics as a way recruit and train undergraduates to do research. The educational and outreach efforts proposed in this project will enable the investigator to motivate students at all levels from diverse backgrounds to use mathematics in tackling some of our most pressing scientific challenges.