Dynamics and Variational Problems

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This research is devoted to problems in mathematical analysisarising from questions in physics. There are six main areas: (1)The ultraviolet problem in quantum electrodynamics, (2) analysis of theprecise 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 ineach case centers on a strategy in which geometric and dynamical analysis of problems in the calculus of variationsplays a key role. We expect, as in our past research, that this strategy willlead us to discover new results and methods in pure analysis, as well as toimprove the understanding of the physical phenomena in question.This research is devoted to problems in mathematical analysisarising from questions in physics. There are five main research areasconcerning a range of problems from mathematically rigorous quantum electrodynamics to hydrodynamics. Ourproposed strategies for researching these topics ineach case centers on a geometric and dynamical analysisof problems in the calculus of variations.We expect, as in our past research, that this strategy willlead us to discover useful new results and methods in pure mathematical analysis, as well as toimprove the understanding of the physical phenomena in question.