Harmonic Analysis and Partial Differential Equations

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This project will continue research activity on the development of various aspects of mathematical analysis which are relevant to some of the most fundamental problems in the area of harmonic analysis and partial differential equations and its applications. The project will also continue the training of graduate students and postdoctoral students to carry out research in these problems. Many of the topics to be studied have their origin in problems coming from physics and engineering. It is expected that the research proposed will have a synergistic effect between these fields and the mathematical fields of analysis and geometry.The main focus of the research will be the study of soliton resolution for the energy critical wave equation and the wave map equation, both in the presence and the absence of symmetry, and for other dispersive and dispersive-geometric models. In another direction, a study will be made of quantitative unique continuation in local settings and at infinity, including the cases of periodic and random coefficients, and their connection with homogenization theory. This is a very ambitious program of research that will have lasting consequences for the development of these areas. The principal investigator will ensure the wide dissemination of the results obtained, including their use in the development of courses, the training of graduate students and postdocs, and the exploration of the connections with other fields of science and technology.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.