Topics in Dispersive Partial Differential Equations and Harmonic Analysis

University Of California-San Diego, La Jolla CA

Investigators

Abstract

This project focuses on two areas of mathematics: namely, partial differential equations and harmonic analysis. Both of them have close connections with the physical world. Every partial differential equation models some physical phenomenon, such as fluid and atmospheric dynamics, particle interactions, behavior of spins in a ferro-magnet, etc. The differential equations that the principal investigator will study in this project are no exception. Harmonic analysis is another area of mathematics with deep implications, such as compressed sensing with applications to signal processing, photography, computed tomography, etc. Although of a varying flavor, all the problems under study in this project have one common feature: trying to understand structure and determining means to measure it. Parts of this project focus on analyzing the long-time dynamics of dispersive partial differential equations. The equations considered in the project have a physical background: the Dirac-type equations and systems model most of the elementary particles and their interactions, while the Schrodinger-maps equation is the Heisenberg model in ferro-magnetism. From a mathematical point of view, the principal investigator is pursuing open problems whose solutions are of broad interest to the scientific community. The long-time dynamics of partial differential equations is a very active field of research that has witnessed some major breakthroughs over the past few years, but in which many fundamental questions have yet to be addressed. Another part of the project focuses on multilinear restriction estimates and their applications. This theory brings together ideas and tools from a mix of fields (e.g., analysis, combinatorics, incidence geometry, differential geometry, algebraic topology). In turn, the multilinear theory has applications to fundamental problems in harmonic analysis, partial differential equations, combinatorics, and number theory.