Harmonic Analysis and Partial Differential Equations

该项目致力于解决长期存在的处理波的非线性传播的数学问题。这些问题的灵感来自科学和工程的重要领域，如水波、激光和非线性光学、铁磁学、粒子物理学和广义相对论。该项目的中心主题是对在计算和实验中观察到的这类问题中出现的大量显著简化的数学研究。反过来，这样的数学研究改进了计算和实验工具，从而产生了重要的新应用。由该协会、他的学生、博士后和合作者开展的研究也被用作研究生和研究生教育培训的基础。作为该项目的一部分，所取得的成果通过出版研究文章和专著以及通过互联网广泛传播。该项目的主要重点是研究能量临界非线性波动方程和波动映射方程在对称性存在和不存在的情况下的孤子解，以及其他色散和色散几何模型的孤子解；研究局部环境和无穷远处的定量唯一延拓，以及与齐化理论有关的周期系数的情况。这是一个研究项目，应该对这些学科的发展产生持久的影响，并建立在首席调查员之前的研究成果的基础上。希望该项目将在上述科学和工程领域与数学分析和几何领域之间产生协同效应，并通过对未来几代研究人员的培训产生重要的教育影响。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The project addresses the solutions of long-standing mathematical problems dealing with non-linear propagation of waves. These problems are inspired by important areas of science and engineering, such as water waves, lasers and nonlinear optics, ferromagnetism, particle physics and general relativity. The central theme of the project is the mathematical study of a remarkable simplification for large time that occurs in such problems, as observed both computationally and experimentally. In turn, such a mathematical study improves the computational and experimental tools, thus yielding important new applications. The research that is developed by the PI, his students, postdocs and collaborators is also used as the basis for educational training at the graduate and postgraduate levels. The results obtained as part of the project are disseminated widely through the publication of research articles and monographs, and through the internet. The main focus of the project is on the study of soliton resolution for the energy critical nonlinear wave equation and the wave map equation, both in the presence and the absence of symmetries, and for other dispersive and dispersive-geometric models; the study of quantitative unique continuation in local settings and at infinity, and in the case of periodic coefficients with connections to homogenization theory. This is a research program that should have lasting consequences for the development of these subjects and which builds on the Principal Investigator's previous research accomplishments. It is hoped that the project will have a synergistic effect between the areas of science and engineering mentioned above and the mathematical fields of analysis and geometry, as well as an important educational impact through the training of future generations of researchers.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.