Final Report Summary - INSTAB12 (Existence of Instabilities in Hamiltonian Systems on lattices and in Hamiltonian Partial differential equations) The IEF Marie Curie fellowship proposal INSTAB12 proposed the analysis of instabilities in different models in dynamical systems. It had two parts. The first one dealt with the phenomenon of growth of Sobolev norms in Hamiltonian PDEs. This problem has drawn considerable attention in the past years since it helps understanding the transfer of energy and it is related to the weak turbulence phenomenon in physics. The second part dealt with the study of instabilities in the Fermi-Pasta-Ulam model, which is a dynamical system on a lattice of finite arbitrarily large dimension. The fellowship was for two years. Nevertheless, since I have obtained a three years postdoctoral fellowship in Universitat Politecnica de Catalunya (Barcelona), I have finished the Marie Curie fellowship five months before the end date. Concerning the first part of the proposal, the problems I wanted to study were the following:1. Growth of Sobolev norms for the cubic wave equation 2. Growth of Sobolev norms for the cubic nonlinear Schrodinger equation (NLS) with a convolution potential3. Unbounded growth of Sobolev norms for the cubic nonlinear Schrodinger equationIn the year and a half of the fellowship, I have obtained several results. I have proved the existence of orbits of the cubic NLS with a convolution potential achieving arbitrarily large (finite) growth of Sobolev norms. I have also obtained time estimates which are analogous to the ones obtained by V. Kaloshin and myself for the cubic NLS with no potential. This result is surprising since it was expected that the potential would stabilize the dynamics and slow down the growth of Sobolev norms. This result has been published in Communications in Mathematical Physics. The first goal I wanted to achieve in this part of the proposal, the study of the cubic wave equation, turned out to be more difficult than expected since the analysis of the associated resonant sets is considerably more difficult than the study of those of the cubic NLS. Therefore, during the fellowship I have worked on problems which can be seen as intermediate steps towards understanding growth of Sobolev norms for the cubic wave equation and other Hamiltonian PDEs. These projects can also be seen as intermediate steps towards studying the unbounded growth (Goal 3) for which I have not obtained results, due to the early termination of the Marie Curie fellowship. One of the problems I have studied has been the growth of Sobolev norms for NLS with other power nonlinearities. I have done this study with Prof. M. Procesi and Dr. E. Haus from University of Rome La Sapienza. We are currently finishing a paper (I attach a preliminary version) where we obtain the same result as in reference  of the proposal for any degree of the nonlinearity. We also obtain time estimates, which are slower than in the cubic case. The importance of this result is twofold. First, the result itself is a considerable generalization of the previous results which shows, as it was expected, that the phenomenon of Sobolev norms is present in much more general settings than the cubic NLS. Second, for this work we have developed techniques which we expect that will shed some light in the future about the growth of Sobolev norms phenomenon for other PDEs and for PDEs defined in other compact manifolds. Indeed, one of the key ingredients in proving the existence of such phenomenon is the understanding of the huge amount of resonant interactions. The difficulty in the analysis of these interactions grows considerably from the cubic case to a general power nonlinearity. In this work, we have developed techniques which can deal with such a general setting. We expect that this analysis will help us later for the analysis, for instance, for the nonlinear wave equation and also for the beam and Hartree equations.One of the classical tools to analyze instabilities in finite dimensional systems (usually called Arnold diffusion) are Normally Hyperbolic Invariant Manifolds (NHIMs). These manifolds are robust under perturbation and they create "roads" which lead to instabilities. NHIMs also exist in infinite dimensional dynamical systems, which may come from PDEs. Jointly with Prof. R. de la Llave (GeorgiaTech) and Dr. Treviño (U. of Tel Aviv and New York University), we have studied the existence and persistence of NHIMs in finite and infinite dimensional dynamical systems We have also considered the weakly hyperbolic case, which appears, for instance, in the nonlinear Schrödinger or wave equations for small data. We are currently finishing a first paper which can be applied to finite dimensional systems and lattices (including the Fermi Pasta Ulam model), and we expect to finish soon a second paper dealing with dispersive PDEs. We expect that the understanding of such NHIMs will help us to construct new paths to large instabilities in PDEs. The second part of the project dealt with the study of instabilities in the Fermi Pasta Ulam model. The plan for the first year was to analyze this model for a small number of particles and the plan for the second year was to generalize the results to an arbitrarily large number of particles. Nevertheless, the first part of the proposal has occupied most of the time of the fellowship and I have had less time to devote to the Fermi Pasta Ulam model. Concerning this second part, I have worked with Prof. V. Kaloshin (U. of Maryland) and Prof. M. Saprykina (U. of Stockholm) on the analysis of the model with a low number of particles. Due to the early termination we have not obtained results but I plan to continue working on this problem after the end of the Marie Curie fellowship. I want to point out that the already mentioned work done with R. de la Llave and R. Treviño also applies to system on lattices like the Fermi-Pasta-Ulam model. The analysis of the normal forms done by Henrici and Kappeler for this model lead to the existence of NHIMs for the full system. In the future I plan to continue analyzing how these invariant objects can create paths which lead to instabilities for the FPU model.