Inverse Problems in Geometry and Partial Differential Equations

The principal investigator will study inverse spectral and scattering theory in three contexts: (1) the spectral theory of nilpotent Lie groups and compact nilmanifolds, (2) the inverse resonance problem for exterior domains and scattering manifolds, and (3) inverse scattering for very singular potentials with applications to nonlinear dispersive equations. In joint work with Ruth Gornet, the principal investigator will attack a fundamental conjecture in inverse spectral theory by studying trace formulas on two-step compact nil- manifolds. Two-step nilpotent Lie groups G and their quotients by co-compact discrete subgroups carry two natural geometric and analytic structures: the Carnot geometry associ- ated with the sub-Laplacian, and the Riemannian geometry associated with the Laplacians for left-invariant metrics. We will study the spectral theory and \geometric optics" associ- ated with the Riemannian Laplacians and sub-Laplacians in order to derive trace formulas for nilmanifolds. Using these trace formulas, the principal investigator and Gornet will study the relationship between length spectrum and Laplace spectrum. The principal investigator will study the geometry of scattering resonances. Scattering resonances are discrete data associated to the wave equation on a complete, non-compact Riemannian manifold or an exterior domain in Rn. Like the eigenvalues of the Laplacian on a compact manifold, they are thought to carry geometric information. To elucidate the geo- metric content of resonances, the principal investigator will continue his work with Carolyn Gordon and Dorothee Schueth to construct pairs and families of non-isometric manifolds and exterior domains with the same scattering resonances, with particular attention to exterior domains in the plane. In joint work with Paul Yang, the principal investigator will consider the inverse resonance problem for conformally Euclidean manifolds. In joint work with Mikhail Shubin, Thomas Kappeler, and Peter Topalov, the princi- pal investigator will also study inverse scattering theory for SchrÄodinger operators on the line with highly singular potentials and its application to well-posedness questions for the Korteweg-de Vries equation. Building on work in progress on the Miura map on the line, the principal investigator and these collaborators will also study well-posedness questions for the modi¯ed Korteweg-de Vries equation. Eigenvalues and scattering resonances are measurable quantities in electromagnetic and acoustic wave propagation. The work carried out here will elucidate the geometric content of eigenvalues and resonances and suggest what geometric quantities can and cannot be recon- structed from such measurements. The inverse scattering method applies to a large family of dispersive equations; the principal investigator's methods used for SchrÄodinger equation on the line with very singular potentials should extend to other inverse scattering prob- lems relevant to this area. The study of spectral theory and trace formulas on nilmanifolds will improve our understanding of analysis and geometry on nilpotent Lie groups and their quotients, and illuminate the connection between spectral and geometric invariants. Grant activities will impact human resources through graduate training and ongoing col- laborations supported by the grant. In related e®orts, the principal investigator and Carolyn Gordon will organize yearly conferences on inverse spectral geometry which will support doc- toral students and recent doctoral graduates; these conferences will bring together experts in scattering theory and di®erential geometry in order to catalyze developments in the inverse resonance problem.