'Multilinear and Nonlinear Harmonic Analysis' :
The scientific purpose of this project is to study three interrelated subjects in harmonic analysis: 1) Wave packet analysis. Initiated by the proof of boundedness of the bilinear Hilbert transform, this type of analysis has been studied intensely and has led to a series of results on multilinear operators and maximal operators. 2) Arithmetic number theory, in particular the study of arithmetic progressions, sum - and difference sets etc. The link between arithmetic number theory and the type of analysis discussed above has recently be reinforced be several results on multilinear operators. Arithmetic number theory is also related to other problems in harmonic analysis such as the famous Kakeya problem. 3) Nonlinear Fourier analysis or scattering theory. Via multilinear expansions of nonlinear operators in scattering theory, the latter is linked to multilinear operators. While the algebraic aspects of scattering theory in one dimension have been widely studied over the past thirty years, a number of basic analytic questions remain open and will be studied in this project.
Wave packet analysis is a discipline in mathematics that can readily be explained to anyone familiar with musical scores. A musical score is the decoding of a complicated musical composition into its most elementary parts, 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 mathematical data into its elementary pieces, each described by the analogue of duration, pitch, and volume. This type of analysis has had a tremendous impact on the way computers deal with large sets of data coming from acoustic signals, images, radar, internet and other telecommunication, etc. This project studies basic mathematical questions associated to wave packet analysis, fundamental research that is likely to improve our understanding not only of the types of processes described above, but also of many different applications within Mathematics as well. This project will also help to maintain an active research group in harmonic analysis at UCLA.
