Harmonic Analysis and Partial Differential Equations

这个提议的目的是继续PI在调和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方程和相关模型的解的孤子分辨率和爆破性质，研究色散几何流的解的全局行为，如波映射和薛定谔映射，研究周期均匀化，研究局部逆问题的唯一性和重建，在最小正则性假设下的椭圆边值问题的研究和数学物理中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的创新研究计划，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研究中的许多主题都起源于来自物理学，工程学和医学成像的问题。此外，要研究的问题汇集在他们的学习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有有关领域。在提案中开发的研究将是研究生课程，迷你课程和PI讲座的基础。将通过出版研究论文、调查文章和专著，通过arXiv（一个可免费获得的预印本电子服务器，主要研究者将其大部分预印本张贴在该服务器上）以及通过主要研究者的网页，广泛传播该报告。此外，这项建议将导致PI的研究生的研究课题。公共研究所积极开展工作，增加代表性不足群体参与研究的机会。在这方面，PI对他将女性数学家培养为博士生和博士后的记录感到非常自豪。PI希望，对这一建议的研究及其广泛传播将导致更多的女研究生和博士后接受PI的培训。

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.