Oscillatory Integrals: Generalized Radon Transforms and Inverse Problems

University Of Rochester, Rochester NY

Investigators

Abstract

Proposal Number: DMS-0138167 PI: Allan Greenleaf ABSTRACT Three problems will be investigated, related by the common thread of oscillatory integrals with degenerate (or singular) phase functions. First, estimates for Fourier integral operators associated with two-sided Morin singularities, both symmetric and asymmetric, will be considered. This will be attempted by using new decompositions of the operators that reflect the underlying geometry of the phase functions. Secondly, global uniqueness will be considered in the Calderon problem for low-regularity conductivities and potentials in dimensions greater than two, using Radon transform decompositions of the potentials. Finally, a detailed microlocal analysis will be made of the problem for low-regularity and/or conormal conductivities in two dimensions. The operators to be studied in this project transform functions on one space to functions on another space (possibly the same, possibly different) in a way in which oscillation (cancellation caused by high-frequency waves) plays an important role. This general type of operator has been central to progress in the last thirty years in the understanding of a wide range of partial differential equations governing, e.g., wave propagation of sound and electromagnetic waves. It turns out that the mathematical analysis of CAT scanners and other noninvasive medical imaging technologies falls under the same framework. The current project should give a better understanding of the mathematical foundations of these techniques. In the longer term, this might contribute to refinement and improvement of these imaging methods.