Dynamics in Geometric Dispersive Equations and the Effects of Trapping, Scattering and Weak Turbulence

One of the major success stories in analysis over the past couple of decades is the deep and detailed insight into the qualitative properties of solutions to nonlinear dispersive PDE from Mathematical Physics which has been gained through application of techniques from harmonic analysis, spectral theory, the calculus of variations, and dynamical systems. In this way, our understanding of the nonlinear waves which characterize the dynamics of various systems in quantum physics, general relativity, optics, and fluid mechanics (just to name a few) has increased enormously over a remarkably short period. This understanding extends to questions of local and global well-posedness, low-regularity solutions, sin-gularity formation, asymptotic behaviour, the existence and stability of special solutions (such as solitons, or various threshold solutions), and the role such special solutions play in the general dynamics. Progress has been such that it could be said, very roughly speaking, that our mathematical comprehension of nonlinear dispersive PDE which are (a) posed on Euclidean space, and (b) of a sub-critical (roughly, conserved quanti-ties provide some natural control over the size of solutions), or even (due to groundbreaking advances of the last few years) critical nature, is now rather good. On the other hand, it is just as reasonable to say that nonlinear wave equations outside of this category are still quite poorly understood, due to inherent new difficulties. For equations posed on compact domain