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Rigorous justification of the wave kinetic theory

The main purpose of this expository note is to give a short account of the recent developments in mathematical wave kinetic theory. After reviewing the physical theory, we explain the importance of the notion of a scaling law, which dictates the relation between the asymptotic parameters as the kinetic limit is taken. This sets some natural limitations on the kinetic approximation that were not precisely understood in the literature as far as we know. We then describe our recent and upcoming works that give the first full, mathematically rigorous, derivation of the wave kinetic theory at the natural kinetic timescale. The key new ingredient is a delicate analysis of the diagrammatic expansion that allows to a) uncover highly elaborate cancellations at arbitrary large order of diagrams, and b) overcome difficulties coming from factorial divergences in the expansion and the criticality of the problem. The results mentioned in this note appear in our recent works [16, 17] as well as an upcoming one in [18].