Harmonic Analysis and Partial Differential Equations

点击翻译按钮获取中文摘要

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.