Singularities in Oscillatory Integrals and Inverse Problems

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ABSTRACT The project involves work on several distinct problems with the commonthemes of controlling, estimating or reconstructing singularities that arise inoscillatory integral operators, Fourier integral operators and inverse problems.The first problem concerns establishing optimal decay rates for oscillatoryintegral operators in three dimensions having generic homogeneous polynomialphase functions. In the second, the Calderon problem for scalar conductivitieswith jump discontinuities in two dimensions will be analyzed, with singularities(suitably interpreted) of boundary data yielding information about thelocations, and possibly sizes, of the jumps of the conductivity in the interior. A linearized version of the identification problem for the attenuated Radontransform will be analyzed using techniques from micro-local and harmonic analysisin the third problem. Finally, a linearized hyperbolic inverse problem involvesstudying the composition of Fourier integral operators associated to specificsingular geometries arising from the presence of caustics occurring genericallyin three dimensions.Progress on these problems will add to the understanding of partial differentialequations, the operators, which are used to solve them, and thebehavior of their solutions. Many laws of nature are expressed as partial differentialequations, which govern physical quantities of interest, such as electromagneticfield strength or pressure exerted by acoustic waves. The techniques to be developedin this project are applicable to equations that govern various kinds of wavepropagation and are based on a geometric point of view in understanding thesingularities, which are present. The last three problems have the potential toassist in the design or refinement of systems for nondestructive testing, medical imaging, and seismic exploration.