Topics in Analysis, Spectral Theory, and Partial Differential Equations

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.