Harmonic Analysis and Dispersive Partial Differential Equations

这个项目集中在两个数学领域：谐波分析和偏微分方程。两者都与我们周围的物质世界有着密切的联系。调和分析是数学的一个领域，特别关注函数和算子的定量性质，这些实体在真实的生活现象的几乎每一个数学模型中都是基本的。谐波分析对信号处理、摄影、计算机断层扫描等许多其他领域都有着深刻的影响。尽管有着不同的风格，但我们在这里解决的问题有一个共同的功能：试图理解结构和测量它的方法。偏微分方程是一个广阔的数学领域，几乎涵盖了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。本计画所要研究的微分方程就是以这些具体的物理模型为基础，部份的研究则集中在多重线性限制估计及其应用。这一理论汇集了思想和工具，从混合领域：分析，组合学，关联几何，微分几何，代数几何和代数拓扑。反过来，多线性理论应用于调和分析、偏微分方程、组合数学和数论中的基本问题。这个项目的其他部分着重于分析色散偏微分方程（PDE）的长时间动力学。在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堡模型。从数学的角度来看，PI提出了一些开放性问题，这些问题的解决方案在PDE社区中引起了广泛的兴趣。偏微分方程的长时间动力学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的突破，而许多根本性的问题还没有得到解决。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.