Problems in Time Frequency Analysis

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This mathematics research project aims at answering questions in harmonic analysis that are of great interest in modern analysis. First, Demeter proposes to investigate the fundamental problem of whether the phase space translates of various classes of functions are linearly independent. Together with his collaborators, Demeter has recently introduced two new tool: One is the spectral theory of random Schroedinger operators, the other one is the theory of simultaneous Diophantine approximation. It is expected that further analysis will reveal connections with various other fields of mathematics. Second, the question of differentiability and boundedness of singular integrals along vector fields has gained a huge momentum over the last five years. Building on his recent work, Demeter proposes to use analytic and combinatorial methods to investigate various Kakeya-like maximal operators, and to obtain optimal bounds for the Hilbert transform along a given number of directions in the plane. Demeter will also investigate the bilinear Fourier restrictions to domains with curvature. Among other things, this involves extending known uniform estimates for the bilinear Hilbert transform. This research project will employ a wide variety of tools from diverse areas of mathematics such as harmonic analysis, combinatorics, number theory, spectral theory and ergodic theory.The publication of the work that is expected to arise from this mathematics research project will enhance the mathematical community's understanding of the important connections among various mathematics research areas, such as harmonic analysis, number theory and ergodic theory. The resolution of the Heil-Ramanathan-Topiwala conjecture could have a significant impact on various sciences, such as signal processing, where Gabor and wavelet analysis play an important role. This research project includes a human resources component consisting of training graduate students and postdoctoral researchers.