Principal Investigator: Victor Guillemin
The principal investigator proposes to study various inverse problems involving the semi-classical Schroedinger operator and, together with Alejandro Uribe and Zuoqin Wang, Zoll-type phenomena for these operators. He also proposes to study, together with Emily Dryden and Rosa Sena-Dias, an equivariant version of the Abreu conjecture: Does the spectrum of the Laplace operator on a toric orbifold determine the orbifold? In addition he proposes to investigate, together with David Jerison, Yves Colin de Verdiere, Steve Zelditch and Hamid Hezari, spectral properties of the wave trace for the Laplace operator on a convex domain in the plane in the vicinity of the perimeter, L. He also proposes to continue his current collaboration with Dan Burns on asymptotic properties of the spectral measures associated with Toeplitz operators on Kahler manifolds and to work with his students on a number of problems in equivariant symplectic geometry: generalizations of the Delzant theorem for Poisson manifolds and manifolds with degenerate symplectic forms and computations in K-cohomology for GKM manifolds. Finally, a long-standing project with Shlomo Sternberg is to write an on-line text book on functorial properties of symplectic manifolds with applications to semi-classical analysis.
The overall theme of this proposal is "inverse problems" in spectral theory of which the most famous classical example is Mark Kac's "Can one hear the shape of a drum." What one "hears" in many of the problems above are discrete sets of spectral data: sophisticated versions of the vibrations of a drum, and what one wants to glean from this data are sophisticated versions of the shape of the drum itself. There has been a lot of progress on problems of this type over the last two decades, but there have been as well a number of cautionary counterexamples which show that further progress may require techniques that are still in their infancy. A large part of the focus of this proposal is developing such techniques.
