The purpose of this proposal is to continue the PI's research on the development of various aspects of harmonic analysis and partial differential equations. The main focus of the PI's research will be the study of soliton resolution and blow-up properties of solutions to the energy critical wave equations and related models, the study of the global behavior of solutions to dispersive geometric flows, such as wave maps and Schrodinger maps, the study of periodic homogenization, the study of uniqueness and reconstruction in local inverse problems, the study of elliptic boundary value problems under minimal regularity assumptions and the study of quantitative unique continuation arising from localization problems in mathematical physics. This is a very ambitious and innovative program of research, which should develop new ideas and tools to treat important problems in the subjects mentioned above, which will have lasting consequences for the developments of these subjects and which builds on the PI's previous research accomplishments.
Many of the topics in the PI's research have their origins in problems coming from physics, engineering and medical imaging. In addition, the problems to be researched bring together in their study tools from different areas of mathematical analysis and geometry, further developing those areas. It is hoped that the synergy thus created will enrich all the fields involved. The research developed in the proposal will be the basis for graduate course, mini-courses and lectures by the PI. It will be disseminated widely through the publication of research papers, survey articles and monographs, through arXiv, a freely available electronic server of preprints, where the PI posts most of his preprints and through the PI's web page. In addition, this proposal will lead to topics of research for the PI's graduate students. The PI works actively to increase the participation in research of underrepresented groups. In this connection, the PI is very proud of his record of training female mathematicians as PhD students and postdocs. The PI hopes that the research in this proposal and its broad dissemination will lead to an even larger number of female graduate students and postdocs to be trained by the PI.
