Dynamics and Variational Problems

这项研究致力于解决由物理问题引起的数学分析问题。有六个主要领域：(1)量子电动力学中的紫外线问题，(2)对动力学理论中Kac型模型接近平衡的精确速率的分析，(3)非平衡统计力学中由相分离问题引起的界面运动问题，(4)尖锐的Lieb-Thirring不等式，(5)寻找简单能量泛函数的最小化模式。我们提出的研究这些不同的问题，在每个情况下，集中在一个策略，其中几何和动态分析的问题，在变分演算发挥关键作用。我们期望，就像我们过去的研究一样，这一策略将引导我们在纯分析中发现新的结果和方法，并提高对所讨论的物理现象的理解。这项研究致力于解决由物理问题引起的数学分析问题。从数学上严格的量子电动力学到流体动力学，有五个主要研究领域涉及一系列问题。我们提出的研究这些主题的策略在每个案例中都集中在变分学问题的几何和动态分析上。我们期望，就像我们过去的研究一样，这一策略将引导我们在纯数学分析中发现有用的新结果和方法，并提高对所讨论的物理现象的理解。

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.