Mathematical Sciences: Harmonic Analysis and Differential Equations

DMS-9706889查尔斯·费弗曼和埃利亚斯·斯坦·费弗曼和斯坦目前正在研究各种问题，涉及调和分析、偏微分方程组和微局部分析技术。它们适用于不同的领域，如量子力学、湍流、伪微分算子、振荡积分、海森堡型群分析和遍历理论。这些问题与以下问题有关：对Thomas-Fermi密度的引导修正的猜想公式；重整化非相对论量子电动力学的稳定性；湍流中的能量级联；振荡积分算子的急剧衰减；乘积结构对次椭圆算子估计的影响；以及半单李群的遍历定理。这个项目的主题是调和分析，它一再证明了它在科学技术中的基本重要性。它是(例如)光学、电磁学的基本工具。量子力学、通信和概率，以及核心数学的一个活跃研究领域。最近，谐波分析在工程上的一个引人注目的应用是小波理论，它现在被用于语音处理和指纹识别等各个领域。

ABSTRACT DMS-9706889 Charles Fefferman and Elias Stein Fefferman and Stein are currently working on a variety of problems, involving techniques of harmonic analysis, partial differential equations, and micro-local analysis. These they apply to such diverse areas as quantum mechanics, turbulence, pseudo-differential operators, oscillatory integrals, analysis on Heisenberg-type groups, and ergodic theory. These problems are connected with such issues as: a conjectured formula for the leading correction to the Thomas-Fermi density; stability of renormalized non-relativistic quantum electrodynamics; energy cascade in a turbulent fluid; sharp decay for oscillatory integral operators; effect of product-structure on estimates for sub-elliptic operators; and ergodic theorems for semi-simple Lie groups. Harmonic analysis, the theme of this project, has repeatedly proven its basic importance in science and technology. It is a fundamental tool in (for instance) optics, electromagnetism. quantum mechanics, communication, and probability, as well as an active research area in core mathematics. A striking recent application of harmonic analysis to engineering is the theory of wavelets, which is now being used in such varied domains as speech processing and fingerprint identification.