An Introduction to Mathematical Wave Kinetic Theory

This course will present the recent progress in the rigorous justification of the wave kinetic theory. This is the theory of non-equilibrium statistical physics for nonlinear waves. The main object to justify is the so-called "wave kinetic equation", which stands as the wave analog of Boltzmann's equation; this is Hilbert's sixth problem for waves. We shall explain this progress in the context of the nonlinear Schrodinger equation, as a representative and universal model for nonlinear dispersive systems. The main methodology is the use of Feynman diagram (or Dyson series) expansions, and the key is to perform a deep analysis of the diagrams that allows to prove the convergence of the expansion. We shall discuss all the new ideas that were developed in this context to resolve this longstanding open problem.