Recent Developments on Certain Dispersive Equations as Infinite Dimensional Hamiltonian Systems

In this talk I will present some recent developments in the study of dispersive differential equations on compact manifolds that may also have the property of being infinite dimensional Hamiltonian systems. I will talk about
Strichartz estimates, weak turbulence, Gibbs measures, symplectic structures and non-squeezing theorems. A list of open problems will conclude the talk.