This project studies time-frequency analysis as it first appeared in the nineteen sixties with Carleson's seminal work on almost everywhere convergence of Fourier series. About fifteen years ago, Carleson's techniques were used to establish estimates for the bilinear Hilbert transform which initiated a rapid development of time-frequency analysis. The research of this project elaborates on important aspects of this theory in small dimensions greater than one with relevance to other areas of analysis. The bilinear Hilbert transform in two dimensions relates to a long standing open problem in ergodic theory about almost everywhere convergence of ergodic averages for two commuting transformations. Time-frequency analysis in two dimensions is also relevant to twin conjectures by Stein and Zygmund on Hilbert transform and differentiation along vector fields in two dimensions. Finally, a non-linear variant of Fourier analysis on low dimensional Lie groups, which relates to many branches of mathematics such as scattering theory, gives rise to interesting fundmental questions. Some of them are nonlinear variants of time frequency analysis in small dimensions.
The research of this project has several connections to mathematical physics, which may lead to applications. It studies the mathematical foundations of scattering of particles along long range potentials. It also studies the foundations of analysis along vector fields, which present the mathematical framework of fluid dynamics. Some of the analysis of this project in two dimensions may be interpreted as a particular interplay of two random processes, such as can be found in financial mathematics. This project will help continue an active research and training group in harmonic analysis at UCLA, that has attracted a number of excellent young mathematicians and has helped maintain the workforce in mathematics. A further educational component of this project is to continue a series of very successful summer schools for graduate students and postdoctoral fellows in analysis from different universities.
