The project involves work on several distinct problems with the common themes of controlling, estimating or reconstructing singularities that arise in oscillatory integral operators, Fourier integral operators and inverse problems. The first problem concerns establishing optimal decay rates for oscillatory integral operators in three dimensions having generic homogeneous polynomial phase functions. In the second, the Calderon problem for scalar conductivities with jump discontinuities in two dimensions will be analyzed, with singularities (suitably interpreted) of boundary data yielding information about the locations, and possibly sizes, of the jumps of the conductivity in the interior. A linearized version of the identification problem for the attenuated Radon transform will be analyzed using techniques from micro-local and harmonic analysis in the third problem. Finally, a linearized hyperbolic inverse problem involves studying the composition of Fourier integral operators associated to specific singular geometries arising from the presence of caustics occurring generically in three dimensions.
Progress on these problems will add to the understanding of partial differential equations, the operators, which are used to solve them, and the behavior of their solutions. Many laws of nature are expressed as partial differential equations, which govern physical quantities of interest, such as electromagnetic field strength or pressure exerted by acoustic waves. The techniques to be developed in this project are applicable to equations that govern various kinds of wave propagation and are based on a geometric point of view in understanding the singularities, which are present. The last three problems have the potential to assist in the design or refinement of systems for nondestructive testing, medical imaging, and seismic exploration.
