Topics in Analysis and Spectral Theory

This project, focused on a range of questions central to classical analysis, spectral theory, and partial differential equations, provides a rigorous study of mathematical models that describe important phenomena in physics, such as the propagation of electromagnetic and acoustic waves. The goal is to develop new analytical tools with a wide scope of applications to other disciplines, such as probability and the theory of random matrices. The Principal Investigator (PI) will train undergraduate, graduate students, and postdocs in research related to the project. In recent years, the PI and his collaborators have made progress in several areas: Steklov's problem in the theory of orthogonal polynomials, the inverse problem for the Szego class on the real line, and the discovery of a connection between the multiple orthogonal polynomials and the spectral theory of Jacobi matrices on trees. This development uncovers the natural directions of research that the PI will pursue in this project. The work will also focus on other topics, such as developing the perturbation technique to better understand electromagnetic scattering in the multidimensional Schrodinger equation. The analysis will blend the methods of harmonic and complex analysis and will be rooted in the physical intuition of the scattering process. Another research direction will focus on studying problems of global well-posedness and asymptotics for classical completely integrable systems using the methods of inverse scattering theory. 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.