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

点击翻译按钮获取中文摘要

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.