Harmonic Analysis and Partial Differential Equations

University Of Chicago, Chicago IL

Investigators

Abstract

The purpose of this project is to advance the principal investigator's research in the development of various aspects of harmonic analysis and partial differential equations. The main foci of the project will be the study of "critical elements" in critical nonlinear and wave equations, the study of universal profiles for blow-up solutions, the study of defocusing energy supercritical wave equations, the development of the connection between issues of uniqueness of solutions to parabolic and dispersive equations with Hardy's uncertainty principle for the Fourier transform, the study of local and global inverse problems, and the development of the theory of homogenization for elliptic equations and systems on Lipschitz domains. Many of the topics under study in this project have their origins in problems coming from physics, engineering, and biotechnology. It is hoped that the proposed research will have a synergistic effect between those fields and the mathematical fields of analysis and geometry. One prominent feature of the proposal is the principal investigator's collaborative research with female mathematicians. The broader impacts of the project reside in the wide dissemination of the results obtained, through publication of articles and monographs and through lectures, courses, and web-sites, in the training of students and postdocs, in the increased participation of underrepresented groups, and in the many connections with other fields of science and technology.