Harmonic Analysis, Structure Theory of Measures, and Properties of Hamiltonian Dynamical Systems

调和分析、结构测度理论以及哈密顿动力系统的性质

• 批准号: 1856124
• 负责人: Bobby Wilson
• 金额: $ 19.64万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1856124&HistoricalAwards=false
• 关键词: Harmonic; Analysis; Structure; Theory; Measures

Harmonic analysis and geometric measure theory are important and fundamental mathematical disciplines to the study of partial differential equations and dynamical systems that arise from differential equations. The aim of this project is to add to the literature of these foundational subjects as well as to the study of a class of partial differential equations that are very important to modeling interaction of particles. Many types of physical phenomena, including the behavior of lasers through various types of media, thermalization of crystals, and optimal control, can be modeled by interacting particles or variational problems -- the two areas of research that benefit most from advances in geometric measure theory and harmonic analysis.The scope of this research project includes the study of structures of measures via the examination of quantitative estimates of geometry, including measures of projections and point-wise densities. At the same time, this project entails the study of Hamiltonian dynamical systems as they relate to nonlinear dispersive PDEs, including nonlinear Schrodinger equations and various water wave equations. Specifically, we will study the question of stability of special solutions and integrability for the cubic nonlinear Schrodinger equation with data defined on compact domains of dimension greater than one. In addition, we will study the decay of the Favard length of neighborhoods of purely unrectifiable sets as well as Besicovitch's one-half problem and Falconer's distance set conjecture.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.

调和分析和几何测度论是研究偏微分方程和由微分方程产生的动力系统的重要基础数学学科。该项目的目的是丰富这些基础学科的文献以及对一类对粒子相互作用建模非常重要的偏微分方程的研究。许多类型的物理现象，包括激光通过各种类型介质的行为、晶体的热化和最优控制，都可以通过相互作用的粒子或变分问题来建模——这两个研究领域最受益于几何测度理论和调和分析的进步。该研究项目的范围包括通过检查几何的定量估计来研究测度结构，包括投影和逐点密度的测度。同时，该项目还需要研究与非线性色散偏微分方程相关的哈密顿动力系统，包括非线性薛定谔方程和各种水波方程。具体来说，我们将研究三次非线性薛定谔方程的特殊解的稳定性和可积性问题，其中数据定义在维度大于一的紧域上。此外，我们还将研究纯粹不可校正集邻域的 Favard 长度的衰减以及贝西科维奇的二分之一问题和 Falconer 距离集猜想。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。