Mathematical Sciences: Harmonic Analysis and Differential Equations

摘要DMS-9706889 查尔斯·费曼和埃利亚斯·斯坦 弗罗曼和斯坦目前正在研究各种 问题，涉及谐波分析技术，部分 微分方程和微观局部分析。 这些他们 适用于量子力学，湍流， 伪微分算子，振荡积分，分析 海森伯型群和遍历理论。 这些问题 与这些问题有关，如： 修正的P-Fermi密度;重整化的稳定性 非相对论性量子电动力学;能量级联 湍流;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; 半单李群的遍历定理。 谐波分析，这个项目的主题，已经多次 证明了它在科学和技术中的重要性。 这是一个 光学和电磁学的基本工具。量子 力学、通信和概率，以及一个活跃的 核心数学研究领域。 最近一个引人注目的应用 谐波分析在工程上的应用最广泛的是小波理论， 现在被用于各种领域，如语音处理和 指纹识别

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.