Some problems in time-frequency analysis

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This project will explore several interconnected questions in the area of modulation-invariant time-frequency analysis. The first group of questions is concerned with the return times phenomenon for measure-preserving systems. Here, one is interested in the convergence of weighted averages along flows in dynamical systems, specifically when the weights are generated by a flow in a second dynamical system. Two problems of particular interest are to determine the amount of Lp regularity required for convergence, especially when the averages are replaced by singular integrals, and to determine, in terms of variation-norm estimates, how quickly this convergence occurs. These problems should be addressed by proving suitable extensions of the Carleson-Hunt theorem. The second group of questions concerns the bilinear Hilbert transform. One problem of interest here is to find new estimates which are uniform in the parameter which determines the singularity set for the transform. A second problem is to determine the rate of convergence for truncated transforms and the associated maximal operator. Although, at first glance, these two groups of questions may not seem to be related, their resolution is expected to be largely guided by a common set of tools. The proposed research falls under the broad umbrella of harmonic analysis, a discipline which studies the decomposition of many types of data into combinations of basic waves of varying frequencies. These decompositions have been broadly applied in areas such as physics, chemistry, biology, and finance. An application which is especially relevant here is data compression, where one seeks to represent a signal, such as a movie or audio track, accurately while using a minimum amount of storage space. Convergence rate estimates, as in the previous paragraph, can in principle be used to characterize how much space is needed to give a "good" representation of a signal. The project also makes significant contact with areas of mathematics outside of harmonic analysis, including ergodic theory and additive combinatorics, and can be expected to stimulate many active discussions with faculty and students of the LSU math department, and with collaborators across the country.