CAREER: Regularity Theory of Measures and Dispersive Partial Differential Equations

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。该项目旨在推进和促进对物理世界理解的基础数学分析的研究。具体来说，这个项目与演化方程相关的部分将提供对玻色气体、非线性光学系统和地球物理流体行为的基本数学模型的更好理解。该项目与几何测量理论相关的部分将增强数学界对用于分析上述类型数学模型的工具和方法的理解。同时，该项目旨在通过为有色人种本科生开展夏季定向阅读项目，对未来分析师的发展采取集中的方法。该项目的活动不仅包括与纯数学研究人员的合作，还包括PI与工程和应用数学研究人员的合作。交叉合作将进一步加强数学科学的实力，同时它为早期职业研究人员提供了发展各种各样的方法来追求数学科学的能力。这个项目的目的是进一步加深我们对几何测量理论基础方面的知识，以及对相互作用的粒子和流体的建模非常重要的偏微分方程的研究。这些研究途径将在未来几年与分析领域保持至关重要的相关性，因此，该项目包括教育活动，将有色人种的学生介绍给这些研究途径。项目的工作遵循两个研究轨道：第一个轨道包括非线性薛定谔方程和Fermi-Pasta-Tsingou-Ulam弹簧质量分子系统等经典色散方程的研究。这条轨道的一个重要组成部分是研究半线性色散演化方程，其色散关系由系统的特定物理方面参数化。第二个轨道包括几何测度理论的研究，特别是Banach空间中的Besicovitch， Marstrand和Preiss的结构理论以及函数的可微性。该项目的目标是确定上述演化方程的解的存在性，并表征这些解的行为，以及表征数学分析研究中自然出现的集合和测量的结构。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). This project seeks to advance and promote the study of mathematical analysis fundamental to the understanding of the physical world. Specifically, the portions of this project related to evolution equations will provide a greater understanding of essential mathematical models of the behavior of Bose gases, nonlinear optical systems, and geophysical fluids. The portion of the project related to geometric measure theory will enhance the mathematical community's understanding of the tools and methods used to analyze mathematical models of the type described above. Simultaneously, the project is designed to take a focused approach towards the development of future analysts by running a summer directed reading program for undergraduate students of color. The project's activities include collaboration with not only researchers in pure mathematics, but a continuation of the PI's work with those in engineering and applied mathematics. The cross-collaboration will further bolster the strength of the mathematical sciences at the same time that it affords early-career researchers the ability to develop a wide variety of ways in which to pursue the mathematical sciences.The aim of this project is to further our knowledge of the foundational aspects of geometric measure theory, as well as study classes of partial differential equations very important to, among other things, the modeling of interacting particles and fluids. These avenues of research will maintain crucial relevance to the field of analysis for years to come and, for this reason, the project includes educational activities that will introduce students of color to these research pathways. The work of the project follows two research tracks: the first track consists of the study of classical dispersive equations such as the nonlinear Schrodinger equation and Fermi-Pasta-Tsingou-Ulam spring-mass molecular system. A significant component of this track is the study of semilinear dispersive evolution equations whose dispersion relations are parametrized by a specific physical aspect of the system. The second track consists of the study of geometric measure theory and, in particular, the structure theory of Besicovitch, Marstrand and Preiss in Banach Spaces as well as differentiability properties of functions. The goal of the project is to determine the existence of solutions to the evolution equations described above and characterize the behavior of such solutions as well as characterize the structure of sets and measures that naturally occur in the study of mathematical analysis.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.