Multilinear and Nonlinear Harmonic Analysis

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ABSTRACTDMS: 0400879Ch Thiele'Multilinear and Nonlinear Harmonic Analysis':The scientific purpose of this project is to studythree interrelated subjects in harmonic analysis:1) Wave packet analysis. Initiated by the proof ofboundedness of the bilinear Hilbert transform, this typeof analysis has been studied intensely and has led toa series of results on multilinear operators and maximaloperators. 2) Arithmetic number theory, in particularthe study of arithmetic progressions, sum - and difference setsetc. The link between arithmetic number theory and thetype of analysis discussed above has recently be reinforcedbe several results on multilinear operators. Arithmetic numbertheory is also related to other problems in harmonic analysissuch as the famous Kakeya problem.3) Nonlinear Fourier analysis or scattering theory. Viamultilinear expansions of nonlinear operators in scatteringtheory, the latter is linked to multilinear operators.While the algebraic aspects of scattering theory in onedimension have been widely studied over the past thirty years,a number of basic analytic questions remain open and willbe studied in this project.Wave packet analysis is a discipline in mathematicsthat can readily be explained to anyone familiar withmusical scores. A musical score is the decoding of acomplicated musical composition into its most elementaryparts, the notes, each described by duration, pitch, and volume.Wave packet analysis is the mathematical analogue of this,which can be used to decode a large variety of mathematicaldata into its elementary pieces, each described by the analogue of duration, pitch,and volume. This type of analysis has had a tremendous impacton the way computers deal with large sets of data coming fromacoustic signals, images, radar, internet and other telecommunication,etc. This project studies basic mathematical questions associated to wave packet analysis, fundamental research that is likely toimprove our understanding not only of the types of processesdescribed above, but also of many different applications withinMathematics as well. This project will also help to maintain anactive research group in harmonic analysis at UCLA.