Space-time resonances

This article is a short exposition of the space-time resonances method. It was introduced by Masmoudi, Shatah, and the author, in order to understand global existence for nonlinear dispersive equations, set in the whole space, and with small data. The idea is to combine the classical concept of resonances, with the feature of dispersive equations: wave packets propagate at a group velocity which depends on their frequency localization. The analytical method which follows from this idea turns out to be a very general tool. This expository article aims at presenting the space-time resonance method, as it is developped by Masmoudi, Shatah, and the author [6] [7] [8], and the author [5].