This award will support mathematical research centering on the construction of scattering operators and the investigation of their properties on compact smooth manifolds with a complete Riemannian metric. This includes topics in dynamical scattering theory on asymptotically hyperbolic manifolds and manifolds with cusps. This allows the scattering matrices to be viewed as an analogue of Dirichlet to Neumann map associated with an operator on a manifold. Another project is to prove certain trace formulae for the spectrum of the Laplacian on asymptotically hyperbolic manifolds and to investigate the distribution of resonances. Certain other properties involving the scattering operator on geometrically finite hyperbolic manifolds will also be studied.
Scattering theory has a long history and was a central topic in mathematical physics. The mathematical analysis of many aspects of the theory has undergone significant advances in recent years and this project will investigate a variety of currently open mathematical questions.
