Mathematical Sciences: Harmonic Analysis and Differential Equations

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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.