CAREER: Regularity Theory of Measures and Dispersive Partial Differential Equations

University Of Washington, Seattle WA

Investigators

Abstract

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). This project seeks to advance and promote the study of mathematical analysis fundamental to the understanding of the physical world. Specifically, the portions of this project related to evolution equations will provide a greater understanding of essential mathematical models of the behavior of Bose gases, nonlinear optical systems, and geophysical fluids. The portion of the project related to geometric measure theory will enhance the mathematical community's understanding of the tools and methods used to analyze mathematical models of the type described above. Simultaneously, the project is designed to take a focused approach towards the development of future analysts by running a summer directed reading program for undergraduate students. The project's activities include collaboration with not only researchers in pure mathematics, but a continuation of the PI's work with those in engineering and applied mathematics. The cross-collaboration will further bolster the strength of the mathematical sciences at the same time that it affords early-career researchers the ability to develop a wide variety of ways in which to pursue the mathematical sciences. The aim of this project is to further our knowledge of the foundational aspects of geometric measure theory, as well as study classes of partial differential equations very important to, among other things, the modeling of interacting particles and fluids. These avenues of research will maintain crucial relevance to the field of analysis for years to come and, for this reason, the project includes educational activities that will introduce students of color to these research pathways. The work of the project follows two research tracks: the first track consists of the study of classical dispersive equations such as the nonlinear Schrodinger equation and Fermi-Pasta-Tsingou-Ulam spring-mass molecular system. A significant component of this track is the study of semilinear dispersive evolution equations whose dispersion relations are parametrized by a specific physical aspect of the system. The second track consists of the study of geometric measure theory and, in particular, the structure theory of Besicovitch, Marstrand and Preiss in Banach Spaces as well as differentiability properties of functions. The goal of the project is to determine the existence of solutions to the evolution equations described above and characterize the behavior of such solutions as well as characterize the structure of sets and measures that naturally occur in the study of mathematical analysis. 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.