Dynamics and Variational Problems

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

This research is devoted to problems in mathematical analysis arising from questions in physics. There are six main areas: (1) The ultraviolet problem in quantum electrodynamics, (2) analysis of the precise rate of approach to equilibrium on models of Kac type for kinetic theory, (3) problems of interface motion arising from phase separation problems in non-equilibrium statistical mechanics, (4) sharp Lieb-Thirring inequalities, and (5) the search for patterns in minimizers of simple energy functionals. Our proposed research into these various problems in each case centers on a strategy in which geometric and dynamical analysis of problems in the calculus of variations plays a key role. We expect, as in our past research, that this strategy will lead us to discover new results and methods in pure analysis, as well as to improve the understanding of the physical phenomena in question. This research is devoted to problems in mathematical analysis arising from questions in physics. There are five main research areas concerning a range of problems from mathematically rigorous quantum electrodynamics to hydrodynamics. Our proposed strategies for researching these topics in each case centers on a geometric and dynamical analysis of problems in the calculus of variations. We expect, as in our past research, that this strategy will lead us to discover useful new results and methods in pure mathematical analysis, as well as to improve the understanding of the physical phenomena in question.