This project involves investigations in Partial Differential Equations, with particular attention to two areas. One involves the development of microlocal methods in spectral and scattering theory. Wave equation techniques continue to be important in the analysis of functional calculi for many important self-adjoint differential operators. One type of problem in this area is the fine behavior of convergence of generalized eigenfunction expansions for the Laplace operator on a broad class of complete Riemannian manifolds. Both in this study and other studies in spectral theory, such as spectral theory for quantum Hamiltonians for gauge fields, there arise problems in the Fourier analysis of Fourier integral distributions associated with transversally intersecting Lagrangians. The project will also treat some inverse scattering problems, in which microlocal techniques for eigenfunction expansions appear as useful tools to filter out that part of the data whose measurement leads to ill posedness. The other general area involves the study of PDE in various nonsmooth settings. Some of the work here has been stimulated by work on equations of fluid motion in nonsmooth domains. This work has also led to other problems, such as applications of layer potential methods to PDE with nonsmooth coefficients, on Lipschitz domains. This work in turn stimulates work on classes of pseudodifferential operators whose symbols are not regular.
The studies of wave motion and fluid motion are fundamental in science and mathematics. Reception of wave motion enables one to decode such motion into information about the source. This may involve hearing a bell ring and inferring its size and location from the tones into which the sound waves resolve. It may involve recording readings of waves generated by a blast in the ground, and drawing conclusions about the ground underneath. The relation between the study of waves produced by a source and the set of tones produced, the `spectrum,' is a central topic in mathematical analysis. This project will further develop mathematical tools to study waves and spectra. It will tackle questions on how diffraction effects of waves influence the reconstruction of functions from their spectra, and questions on how to reconstruct an object from the waves it scatters. It will also tackle the equations of fluid motion, using techniques of harmonic analysis that are also germane to the study of waves.
