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Harmonic Analysis and Partial Differential Equations

调和分析和偏微分方程

基本信息

批准号：
1800082
负责人：
Carlos Kenig
金额：
$ 25.24万
依托单位：
University of Chicago
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-09-01 至 2021-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800082&HistoricalAwards=false
关键词：
Harmonic Analysis Partial Differential Equations

项目摘要

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.

该项目将继续研究数学分析各个方面的发展，这些方面与调和分析和偏微分方程及其应用领域的一些最基本问题有关。该项目还将继续培训研究生和博士后研究生，以便对这些问题进行研究。许多要研究的主题都起源于物理学和工程学的问题。预计所提出的研究将在这些领域与分析和几何的数学领域之间产生协同效应。研究的主要重点将是研究在存在和不存在对称性的情况下能量临界波动方程和波图方程以及其他色散和色散几何模型的孤子分辨率。在另一个方向上，将研究在局部设置和无穷远处的定量唯一连续性，包括周期和随机系数的情况，以及它们与均匀化理论的联系。这是一个非常雄心勃勃的研究计划，将对这些领域的发展产生持久的影响。主要研究者将确保所获得的结果的广泛传播，包括它们在课程开发中的使用，研究生和博士后的培训，以及与其他科学和技术领域的联系的探索。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。