Oscillatory Integral Operators, Inverse Problems and Non-Transformation Optics

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This proposal consists of four groups of problems. Three of them are concerned with the interaction between geometry and analysis, the study of functions and the properties of operators, which are transformations that turn functions into new functions. The operators studied in this project arise either when trying to understand how waves propagate (for example, acoustic waves in the Earth), or in basic mathematical tools that are used to study wave propagation, or in the study of geometric properties of large sets of points. Waves can be idealized as traveling along rays, such as light rays, and when the rays concentrate in a small region, new methods need to be developed to obtain accurate descriptions of the waves. Two parts of this project will continue the principal investigator's study of various operators and how their properties can be predicted from the structure of rays or more general underlying geometry. Possible applications include improved understanding of artifacts in seismic imaging. A third part concerns multilinear operators, which act on several functions at a time. These arise in the study of how two or more waves interact and also in the study of point clouds in discrete geometry. The fourth part is a new approach to the design of resonant devices, such as antennas. Recent progress in materials science, physics, and mathematics has led to rapid advances in the design of devices constructed from structured composite materials, also called metamaterials, which have radical effects on wave propagation. One particularly successful approach is based on transformation optics. The principal investigator will investigate a new, non-transformation optics design methodology. Two of the proposed projects concern developing new tools for dealing with degeneracies or singularities of smooth or real-analytic functions. In one, composition of degenerate Fourier integral operators with smooth phases leads to operators that have wave-front relations that are not smooth, and understanding how to associate Fourier-integral-operator-like operators to these geometries will expand the reach of microlocal analysis and help analyze certain inverse problems. In another, new techniques will need to be found to deal with oscillatory integral operators with real-analytic phases in one plus two or two plus two dimensions. The difficulties include understanding to what extent two or more functions can have their zero varieties simultaneously resolved. Some of the same techniques will also be applied to try to find an algorithmic description of jumping numbers, which are the subject of current interest in singularity theory and algebraic geometry. Doing so will strengthen the connection between analysis and these fields. Progress on the third part of the project will not only further develop the theory of multilinear operators within harmonic analysis, but it will also have immediate applications to geometric measure theory and discrete geometry. The final component will show how established ideas from linear partial differential equations can be useful in designing and rigorously verifying the properties of resonant structures such as antennas and will help contribute a new design methodology to the rapidly developing area of metamaterials.