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

调和分析和几何测度论是研究偏微分方程和由微分方程导出的动力系统的重要基础数学学科。这个项目的目的是增加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，包括激光在不同类型介质中的行为，晶体的热化和最优控制，都可以用相互作用的粒子或变分问题来模拟--这两个研究领域从几何测量理论和调和分析的进步中受益最大。这个研究项目的范围包括通过检查几何的定量估计来研究测量的结构，包括投影和逐点密度的测量。同时，这个项目需要研究哈密顿动力系统，因为它们与非线性色散偏微分方程组有关，包括非线性薛定谔方程和各种水波方程。具体地说，我们将研究定义在一维以上紧致区域上的三次非线性薛定谔方程特解的稳定性和可积性问题。此外，我们将研究纯不可纠集邻域的Favard长度的衰减以及Besicovitch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优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.