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

这个项目涉及对流体流动模型的数学基础系统的研究。这些方程的应用包括血浆在血液流动中的运动，石油在管道中的运动，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，可以用来建立非线性偏微分方程系统解的整体存在性，这些系统在各种物理现象中都主张质量和动量守恒。除了让学生参与研究之外，研究人员还将试点一个“通往博士的桥梁”项目，旨在增加宾夕法尼亚大学数学专业女性和少数族裔获得博士学位的人数。研究人员还将与费城工场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与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欧拉-泊松方程，一个用于研究连续质量模型材料行为的偏微分方程系统。这些方程作为一个作用在概率测度空间中的临界点出现；该项目将调查最小化这一行动存在的充分和必要条件。此外，研究者的目的是为拉格朗日力学理论、变分法和概率测度空间控制论奠定基础。为此，他将运用流体力学、最优输运、函数不等式和进化方程等方法。研究者将与宾夕法尼亚大学教学与学习中心合作，组织一个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。在这个项目中提出的教育和推广工作将使研究者能够激励来自不同背景的各级学生使用数学来解决我们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.