Topics in Dispersive Partial Differential Equations and Harmonic Analysis

这个项目着重于数学的两个领域：即偏微分方程和谐波分析。两者都与物理世界有着密切的联系。每一个偏微分方程都模拟了一些物理现象，如流体和大气动力学、粒子相互作用、铁磁体中的自旋行为等。在这个项目中，首席研究员将要研究的微分方程也不例外。谐波分析是另一个具有深远影响的数学领域，如应用于信号处理、摄影、计算机断层扫描等的压缩感知。尽管各不相同，但本项目所研究的所有问题都有一个共同的特点：试图理解结构并确定测量它的方法。这个项目的部分重点是分析色散偏微分方程的长时间动力学。项目中考虑的方程有物理背景：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中的海森堡模型。从数学的角度来看，首席研究者追求的是开放问题，其解决方案对科学界具有广泛的兴趣。偏微分方程的长时动力学是一个非常活跃的研究领域，在过去几年中取得了一些重大突破，但其中许多基本问题尚未得到解决。项目的另一部分侧重于多线性限制估计及其应用。这一理论汇集了来自不同领域的思想和工具（例如，分析、组合学、关联几何、微分几何、代数拓扑）。反过来，多元线性理论在调和分析、偏微分方程、组合学和数论等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.