I plan on working on the following specific problems. First I would like to prove new Strichartz estimates without using explicit parametrices. This could potentially greatly widen the scope of their applications. I would also like to prove almost global existence for quasilinear wave equations and investigate whether there is global existence for nonlinear wave equations outside of nontrapping obstacles. I would like to use the techniques developed on these projects to study problems in relativity theory, including the stability of the Schwarzschild solution. Finally, I would like to continue my work on problems in harmonic analysis that arise in geometrical situations.
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.
