Harmonic Analysis and Partial Differential Equations

本提案的目的是继续PI在谐波分析和偏微分方程的各个方面的发展的研究。PI的主要研究重点将是研究能量临界波动方程和相关模型解的孤子解析和爆破性质，研究色散几何流（如波图和薛定谔图）解的全局行为，研究周期均匀化，研究局部逆问题的唯一性和重构。最小正则性假设下椭圆边值问题的研究及数学物理中由局部化问题引起的定量唯一延拓的研究。这是一个非常雄心勃勃和创新的研究项目，它应该开发新的想法和工具来处理上述主题中的重要问题，这将对这些主题的发展产生持久的影响，并建立在PI以前的研究成果之上。PI的许多研究课题都源于物理学、工程学和医学成像领域的问题。此外，要研究的问题汇集在他们的研究工具从数学分析和几何的不同领域，进一步发展这些领域。希望由此产生的协同作用将丰富所涉及的所有领域。在提案中进行的研究将成为PI的研究生课程、迷你课程和讲座的基础。它将通过出版研究论文、调查文章和专著、通过免费提供的预印本电子服务器arXiv和通过PI的网页广泛传播。PI在arXiv上发布了他的大部分预印本。此外，该提案将为PI的研究生提供研究课题。PI积极开展工作，增加代表性不足群体对研究的参与。在这方面，PI对他培养女数学家博士研究生和博士后的记录感到非常自豪。该计划希望这项提案中的研究及其广泛传播将导致该计划培养更多的女研究生和博士后。

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.