This talk is devoted to the study of Schrödinger equations in the presence of resonant interactions that can lead to energy transfer. When the domain is a Diophantine torus we prove that, over very long time scales, the majority of small solutions in high regularity Sobolev spaces do not exchange energy from low to high frequencies. We first provide context on Birkhoff normal form approaches to study of the long-time dynamics of the solutions to Hamiltonian partial differential equations. Then, we introduce the induction on scales normal form, central to our proof. Throughout the iteration, we ensure appropriate non-resonance properties while modulating the frequencies (of the linearized system) with the amplitude of the Fourier coefficients of the initial data. Our main challenge is then to address very small divisor problems and to describe the set of admissible initial data. The results are based on a joint work with Joackim Bernier, and an ongoing joint work with Gigliola Staffilani.

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