Topics in Fourier Analysis

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Proposal: DMS-9970042Principal Investigator: Andreas SeegerAbstract: The proposed research will be concerned with a number of topics in Fourier analysis related to singular integral operators, Fourier integral operators with degenerate canonical relations, Radon transforms, averages over curves and surfaces and associated maximal functions, regularity of wave propagation, spectral multipliers on noncompact manifolds, and the multitude of interrelations between these topics.The subject of Fourier analysis has provided powerful tools for various areas of mathematics and its applications. The P.I. proposes to develop further and refine some of these tools. This should lead to a better qualitative understanding of integral operators that arise in the study of partial differential equations from mathematical physics, as well as in the study of integral geometry and tomography. The potential concrete implications of improving this understanding are enormous. Tomography, for instance, not only provides the mathematical foundation for the reconstruction methods that are employed constantly in medical imaging but also furnishes the key to interpreting seismic data, lending it a "real world" significance that cannot be overstated.