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

职业：概率测度空间中的行动最小化路径

• 批准号: 1554130
• 负责人: Ryan Hynd
• 金额: $ 44.99万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-01 至 2022-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1554130&HistoricalAwards=false
• 关键词: CAREER; Action; Minimizing; Paths; Space

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.

这个项目涉及到对流体流动模型的数学基础方程系统的研究。 这些方程的应用包括血液流动中血浆的运动，石油在管道中的流动，以及木星和土星等气态行星的形成。该研究的目的是开发方法，可用于建立非线性偏微分方程系统的整体存在的解决方案，断言在各种物理现象的质量和动量守恒。 除了让学生参与研究外，研究人员还将试行一项“通往博士学位的桥梁”计划，旨在增加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。研究人员还将与费城车间学校的教育工作者合作，将数学建模引入他们的课程，并培训学生团队参加一年一度的穆迪大型数学挑战赛，该比赛由工业和应用数学学会组织。建议的研究重点是基于广义欧拉-泊松方程，一个系统的偏微分方程用于研究的行为的材料建模为连续质量。这些方程出现在概率测度空间中作为行动的临界点;该项目将研究该行动存在极小的充分和必要条件。此外，调查员的目的是奠定基础的拉格朗日力学，变分法，控制理论的概率措施的空间理论。为此，他将采用流体力学，最佳运输，功能不等式和演化方程的方法。与宾夕法尼亚大学的教学中心合作，研究人员将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人员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。