Lie and his collaborators will investigate a number of problems that lie at the interface of time-frequency analysis and other areas of mathematics such as additive combinatorics, geometric measure theory, incidence geometry and partial differential equations. The main objectives for the time-frequency analysis-themed problems in this project are (a), to develop a satisfactory generalized wave packet theory and (b), to develop the proper environment for the time - frequency analysis in higher dimensions. One step in the first direction was taken by the PI in his Ph.D. thesis, where he introduced a new perspective on the generalized wave packet theory which later proved beneficial in proving the boundedness of the Polynomial Carleson operator in dimension one. Some of these techniques appear to be applicable to the study of the boundedness of the maximal Schrodinger operator in dimension one. The PI and his collaborators will investigate several other problems, including: boundedness of the Bilinear Hilbert transform along curves; boundedness of the higher dimensional version of the Polynomial Carleson operator; pointwise convergence of the solution to the free Schrodinger equation in higher dimensions; convergence of the subsequences of sequences of partial Fourier sums.
This mathematics research project investigates various problems in harmonic analysis, which is an area of mathematics that has important applications to problems that arise in other sciences, such as pattern recognition in computer graphics; electrical charges distribution in physics; wave propagation in seismology. The techniques to be employed will bridge several areas of mathematics, including partial differential equations, incidence geometry, and additive combinatorics. The resulting synergies will enrich the arsenal of analytical tools and available techniques; these will be presented in graduate seminars designed with the specific purpose to attract students and young researchers to this dynamic area of mathematics and its applications.