This proposal is concerned with estimates of wave equations on (both compact and non-compact) Riemannian manifolds, possibly with boundary. We are 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 Lebesgues-space estimates (both linear and bilinear) in space for eigenfunctions and quasi-modes, and (local or global) Strichartz estimates in space-time. 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 quasi-modes in spectral asymptotics. In the non-compact setting it is also closely related to the distribution of resonances and their relations to trapped geodesics.
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 I am especially interested in making further contributions to this area.
My NSF award supported my research and also the research of my students and postdocs. I mainly studied nonlinear wave equations and eigenfunctions on Riemannian manifolds. Nonlinear wave equations play an important role in physics, especially in relativity theory. With my NSF funding I was able to establish several new cases where such nonlinear equations can be solved. I worked with researchers throughout the world and we were able to develop new techniques to study these equations, which especially applied to the situation where there is a boundary (e.g. the coast for water waves). Primarily we developed new estimates which go by the name "Strichartz estimates" in mathematical analysis. I also worked extensively on related problems concerning eigenfunctions. These correspond to fundamental modes of vibrations in physical systems and their study represents "global harmonic analysis". In collaboration with Hart Smith, we were able to obtain the first sharp estimates for eigenfunctions in the very physically relevant, but difficult, situation where there is a boundary. Using these estimates we could prove new results in harmonic analysis on Riemannian manifolds. We also were able to adapt the proofs to obtain important new Strichartz estimates that allowed ourselves and other researchers to obtain the first existence results for critical wave equations in domains. Zelditch and I were also able to improve new results concerning the nodal sets of eigenfunctions, which allowed significant new progress on a conjecture of Yau.
