This proposal deals with four distinct problems having the unifying theme of controlling, estimating or reconstructing singularities that arise in oscillatory and Fourier integral operators and inverse problems, as well as the opposite goal of using singularities to obtain new designs in the emerging field of transformation optics. The first problem concerns optimal decay estimates for oscillatory integral operators in two dimensions. In the second, degenerate versions of the Carleson-Sjolin estimates will be pursued. The third project involves a linearized inverse problem for wave equations arising in seismic imaging which leads to analysis and composition of degenerate Fourier integral operators. Finally, analysis on singular spaces will be used to understand the theoretical limits to cloaking and other transformation optics designs.
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, pressure exerted by acoustic waves or the quantum-mechanical probability that a particle will be found in a particular place. 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 and manipulating the singularities which are present. In particular, the third project has the potential to facilitate seismic exploration in situations where images are distorted by the presence of multiple rays connecting seismic source to receiver. The fourth project concerns theoretical underpinnings of the recently emergent field of transformation optics, which has the potential to produce unusual effects on various kinds of waves (e.g., electromagnetic, acoustic, quantum mechanical) using artificially structured materials known as metamaterials.
