Explore a 46-minute lecture by Michela Procesi on stability and recursive solutions in Hamiltonian Partial Differential Equations (PDEs) on compact manifolds. Delve into the existence of special recursive solutions near elliptic fixed points, examining their stability and instability properties. Investigate the complex interplay between chaotic and recursive phenomena caused by resonances and small divisors, utilizing KAM theory methods. Focus on stability properties near fixed points and the existence and stability of quasi-periodic and almost-periodic solutions. Cover topics including non-linear PDEs, dynamical systems, invariant tori, perturbation theory, small solutions, linear theory, KAM in infinite dimensions, and reversible autonomous NLS equations. Examine generic tangential sites, combinatorial theorems, and finite regularity solutions for NLS, concluding with open problems in the field.