Long time dynamics of Schr\"odinger and wave equations on flat tori

We consider a class of linear time dependent Schr\"odinger equations and quasi-periodically forced nonlinear Hamiltonian wave/Klein Gordon and Schr\"odinger equations on arbitrary flat tori. For the linear Schr\"odinger equation, we prove a $t^\epsilon$ $(\forall \epsilon >0)$ upper bound for the growth of the Sobolev norms as the time goes to infinity. For the nonlinear Hamiltonian PDEs we construct families of time quasi-periodic solutions. Both results are based on "clusterization properties" of the eigenvalues of the Laplacian on a flat torus and on suitable "separation properties" of the singular sites of Schr\"odinger and wave operators, which are integers, in space-time Fourier lattice, close to a cone or a paraboloid. Thanks to these properties we are able to apply Delort abstract theorem [Del10] to control the speed of growth of the Sobolev norms, and Berti-Corsi-Procesi abstract Nash-Moser theorem [BCP15] to construct quasi-periodic solutions.