Topics in harmonic analysis

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Recent years have seen a burst of activity in the interface of harmonic analysis and its related fields (such as partial differential equations, combinatorics, number theory, and mathematical physics), with beautiful andmutually beneficial consequences. Long-standing open problems in harmonic analysis have been solved using combinatorial techniques, while tools from harmonic analysis have helped in proving exciting results in number theory, partial differential equations and mathematical physics. In this proposal, the applicant outlines future goals in three areas of harmonic analysis: (a) Fourier estimates for cones and their applications, (b) analysis of linear and monomial polyhedra, and (c) oscillatory integral operators with degenerate phases.Projects 1 and 2 address a class of problems that may be approached using Wolff's microlocal estimate forlight cones (arising in his proof of the local smoothing conjecture), and would answer a variety of questionsregarding averaging and Fourier integral operators and partial differential equations. They necessitate a deeper understanding and extension of the delicate combinatorial arguments employed in his proof. Results from project 3 would mark nontrivial progress on a much-studied open question on Bergman kernel estimates, and would be of significant interest in several complex variables. Project 4 addresses a problem of which verylittle is known at the moment, but which has deep ties not only to numerous questions in harmonic analysis (regarding degenerate Fourier integral operators), but also to the extremely rich theory of resolution of singularities.