Harmonic Analysis and Dispersive Partial Differential Equations

这个项目关注数学的两个领域：调和分析和偏微分方程式。它们都与我们周围的物理世界有着密切的联系。调和分析是数学的一个领域，它特别关注函数和运算符的定量性质，这些实体在现实生活现象的几乎每一个数学模型中都是基本的。调和分析对信号处理、摄影、计算机断层成像等许多其他领域都有深刻的影响。尽管不同，我们在这里解决的问题有一个共同的特点：试图理解结构和测量它的方法。偏微分方程是一个广泛的数学领域，涵盖了几乎所有来自物理学的数学模型，从基本的牛顿力学到流体和大气动力学，再到粒子相互作用。我们在这个项目中要研究的微分方程就是基于这样的具体物理模型。这个项目的部分内容集中在多线性约束估计和应用上。这一理论汇集了多个领域的思想和工具：分析、组合学、关联几何、微分几何、代数几何和代数拓扑学。反过来，多线性理论也应用于调和分析、偏微分方程组、组合学和数论中的基本问题。本项目的其他部分侧重于分析色散偏微分方程组的长时间动力学。这个项目中考虑的方程有一个物理背景：Dirac-Klein-Gordon系统模拟了大多数基本粒子及其相互作用，薛定谔映射方程是铁磁性中的海森伯格模型。从数学的角度来看，PI正在提出公开的问题，为这些问题寻找解决方案是PDE社区广泛关注的。PDE的长期动态是一个非常活跃的数学领域，在过去的几年里已经取得了一些重大突破，但许多基本问题还没有解决。这个奖项反映了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。

This project focuses on two areas of Mathematics: Harmonic Analysis and Partial Differential Equations. Both of them have close connections with the physical world around us. Harmonic Analysis is an area of Mathematics that focuses in particular on quantitative properties of functions and operators, entities that are fundamental in almost every mathematical model of a real life phenomena. Harmonic Analysis has deep implications to many other areas such as signal processing, photography, computed tomography, etc. Although of a different flavor, the problems we address here have one common feature: trying to understand structure and means of measure it. Partial Differential Equations is a vast area of mathematics that covers almost all mathematical models coming from Physics, from basic Newtonian mechanics to fluid and atmospheric dynamics to particle interactions. The Differential Equations we propose to study in this project are based on such concrete physical models.Parts of this project focus on the multilinear restriction estimate and applications. This theory brings together ideas and tools from a mix of fields: Analysis, Combinatorics, Incidence Geometry, Differential Geometry, Algebraic Geometry and Algebraic Topology. In turn, the multilinear theory has applications to fundamental problems in Harmonic Analysis, PDE's, Combinatorics and Number Theory. Other parts of this project focus on analyzing the long time dynamics of dispersive Partial Differential Equations (PDE's). The equations considered in this project have a physical background: the Dirac-Klein-Gordon system models most of the elementary particles and their interactions, and the Schroedinger Maps equation is the Heisenberg model in ferro-magnetism. From a mathematical point of view, the PI is proposing open problems for which finding solutions is of broad interest in the PDE community. The long time dynamics of PDE's is a very active field of mathematics, where some major breakthroughs have been achieved over the past few years, while many fundamental problems have yet to be addressed.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.