Harmonic Analysis and Partial Differential Equations

这个项目的目的是推进主要研究者在谐波分析和偏微分方程的各个方面的发展的研究。该项目的主要重点将是研究临界非线性和波动方程中的“临界元素”，研究爆破解的通用剖面，研究散焦能量超临界波动方程，发展抛物方程和色散方程解的唯一性问题与哈代的傅里叶变换不确定性原理之间的联系，研究局部和全局逆问题，以及Lipschitz域上椭圆型方程和方程组的均匀化理论的发展。本项目研究的许多主题都起源于物理学、工程学和生物技术领域的问题。人们希望，拟议的研究将有一个协同效应之间的这些领域和数学领域的分析和几何。该提案的一个突出特点是主要研究者与女性数学家的合作研究。该项目更广泛的影响在于通过发表文章和专著以及通过讲座、课程和网站广泛传播所取得的成果，培训学生和博士后，增加代表性不足的群体的参与，以及与其他科学和技术领域的许多联系。

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.