1Periodic Schroödinger equations in Hamiltonian form

In these lectures I will summarize some old and recent results concerning dif-ferent aspects of periodic Schrödinger equations viewed as infinite dimension Hamiltonian systems. In the first lecture I will recall few facts about dispersive equations and nonlinear Schrd́ingier equations in general, see Sections 1.1 and 1.2. In the second lecture, see Section 1.3, I will start with the classical Strichartz in-equalities in one and two dimensions due to Bourgain. I will continue with some results of local well-posedness for certain nonlinear Schrödinger equa-tions. In the third lecture, see Section 1.4, I will elaborate on the growth in time of high order Sobolev norms for global smooth solutions. I will explain how the estimate of this growth could give some information on how the frequency profile of a certain wave solution could move from low to high frequencies while maintaining constant mass and energy (forward cascade.) I will present two results for the defocusing, cubic, periodic, two dimensional Schrödinge