Topics in Dispersive Partial Differential Equations and Harmonic Analysis

该项目侧重于两个数学领域：即偏微分方程和调和分析。两者都与物质世界有着密切的联系。每一个偏微分方程模型的一些物理现象，如流体和大气动力学，粒子相互作用，自旋在铁磁体的行为，等等的微分方程，首席研究员将在这个项目中研究也不例外。谐波分析是另一个具有深刻影响的数学领域，例如压缩传感与信号处理，摄影，计算机断层扫描等的应用。尽管风格各异，但本项目研究的所有问题都有一个共同的功能：试图理解结构并确定测量方法。本项目的部分重点是分析色散偏微分方程的长期动力学。该项目中考虑的方程具有物理背景：狄拉克型方程和系统模拟了大多数基本粒子及其相互作用，而薛定谔映射方程是铁磁性中的海森堡模型。从数学的角度来看，首席研究员正在追求开放的问题，其解决方案是科学界广泛感兴趣的。偏微分方程的长时间动力学是一个非常活跃的研究领域，在过去的几年里取得了一些重大突破，但其中许多基本问题尚未得到解决。该项目的另一部分侧重于多线性约束估计及其应用。这一理论汇集了来自不同领域的想法和工具（例如，分析，组合学，关联几何，微分几何，代数拓扑）。反过来，多重线性理论在调和分析、偏微分方程、组合学和数论中的基本问题上也有应用。

This project focuses on two areas of mathematics: namely, partial differential equations and harmonic analysis. Both of them have close connections with the physical world. Every partial differential equation models some physical phenomenon, such as fluid and atmospheric dynamics, particle interactions, behavior of spins in a ferro-magnet, etc. The differential equations that the principal investigator will study in this project are no exception. Harmonic analysis is another area of mathematics with deep implications, such as compressed sensing with applications to signal processing, photography, computed tomography, etc. Although of a varying flavor, all the problems under study in this project have one common feature: trying to understand structure and determining means to measure it. Parts of this project focus on analyzing the long-time dynamics of dispersive partial differential equations. The equations considered in the project have a physical background: the Dirac-type equations and systems model most of the elementary particles and their interactions, while the Schrodinger-maps equation is the Heisenberg model in ferro-magnetism. From a mathematical point of view, the principal investigator is pursuing open problems whose solutions are of broad interest to the scientific community. The long-time dynamics of partial differential equations is a very active field of research that has witnessed some major breakthroughs over the past few years, but in which many fundamental questions have yet to be addressed. Another part of the project focuses on multilinear restriction estimates and their applications. This theory brings together ideas and tools from a mix of fields (e.g., analysis, combinatorics, incidence geometry, differential geometry, algebraic topology). In turn, the multilinear theory has applications to fundamental problems in harmonic analysis, partial differential equations, combinatorics, and number theory.