Propagation of waves on domains with boundary

We are concerned with localization properties of solutions to hyperbolic PDEs, especially problems with a geometric component: how do boundaries influence spreading and concentration of solutions. Our first focus is on wave and Schrödinger equations on manifolds with boundary, but strong connections exist with phase space localization for (clusters of) eigenfunctions, which are of independent interest. Motivations come from nonlinear dispersive models (in physically relevant settings), properties of eigenfunctions in quantum chaos, or harmonic analysis on manifolds.