This research is concerned with estimates of wave equations on (both compact and noncompact) Riemannian manifolds, possibly with boundary. The investigator is interested in how the geometry, the boundary and the regularity of the metric influence certain basic estimates. Problems of this kind arise in the study of harmonic analysis on manifolds, the study of local and global solutions of nonlinear wave equations and in the study of eigenfunctions in quantum chaos. Although these topics are widely separated in their physical and historical origins, the relevant mathematics is closely related. Techniques and insights in the various areas cross-fertilize each other in a fruitful way. In particular, a common theme of much current research (and the problems in this proposal) is to try to understand and exploit the mass concentration of eigenfunctions and solutions of linear and nonlinear wave equations. The basic estimates that we have in mind are Lebesgue-space estimates (both linear and bilinear) in space for eigenfunctions and quasimodes, and (local or global) Strichartz estimates in spacetime. The main questions center around how the geometry and especially the presence of a boundary affects the estimates and the kinds of solutions that saturate them. The latter issue is closely related to the much studied (but still not well understood) questions of concentration, oscillation and size properties of modes and quasimodes in spectral asymptotics. In the non-compact setting it is also closely related to the distribution of resonances and their relations to trapped geodesics. In the compact setting we are also interested in exploring how concentration properties of eigenfunctions are related to their nodal sets, which is the set of points where the function vanishes. The investigator wishes to use current and new estimates for eigenfunctions to make further progress on Yau's conjecture about the size of these sets. He also is interested in seeing how certain curvature assumptions affect these estimates and exploring connections with quantum ergodicity.
The above problems arise naturally from interactions between mathematics and areas in physics that include general relativity, quantum mechanics, and quantum chaos. The techniques employed include stationary phase and the study of propagation of singularities. There is a very active group of researchers in quantum physics groups at major universities studying high-energy eigenstates, and the investigator is especially interested in making further contributions to this area.
