Topics in Analysis, Spectral Theory, and Partial Differential Equations

这个研究项目涵盖了一系列经典分析、谱理论和偏微分方程的核心问题。项目重点是为一些重要的物理现象，如电磁波和声波在粗糙或随机介质中的传播，建立严格的数学模型。目标是开发具有一系列应用的分析工具，包括概率论和随机过程理论领域。该项目将使用谐波和复杂分析的工具，包括波包分解、奇异积分算子的加权估计和时频分析。指导学生和初级研究人员将是该项目不可或缺的一部分。本课题将研究几个经典正交系统的性质及其产生的算子。这包括研究不同类型测度的正交多项式的大小，理解树上Jacobi矩阵的谱理论与多重正交概念之间的联系，以及将一维结果推广到多重正交的情况。该项目的一个重要部分将集中在发展德布朗日正则系统和克莱因弦的Szegö理论，并可能将这些结果应用于非线性演化方程。本文还将研究高维椭圆方程的散射理论，以及线性演化方程中能量级联的相关问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This research project covers a range of questions central to classical analysis, spectral theory, and partial differential equations. The project focuses on rigorous underpinning of mathematical models for some essential phenomena in physics, such as electromagnetic and acoustic wave propagation in rough or random media. The goal is to develop analytical tools with a range of applications, including the fields of probability and the theory of stochastic processes. The project will employ tools from harmonic and complex analysis including wave packet decomposition, weighted estimates for singular integral operators, and time-frequency analysis. Mentoring students and junior researchers will be an integral part of the project. This project will investigate the properties of several classical orthogonal systems and the operators generated by them. This includes studying the size of orthogonal polynomials for different types of measures, understanding the connection between spectral theory of Jacobi matrices on trees and the concept of multiple orthogonality, and generalization of one-dimensional results to the case of multiple orthogonality. A significant part of the project will focus on developing the Szegö theory for de Branges canonical systems and Krein strings and possible application of these results to nonlinear evolution equations. Scattering theory for elliptic equations in higher dimensions will also be investigated, and related questions of energy cascade in linear evolution equations will be considered.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.