KdV and Almost Conservation Laws

This short survey paper is concerned with a new method to prove global well-posedness results for dispersive equations below energy spaces, namely $H^1$ for the Schr"odinger equation and $L^2$ for the KdV equation. The main ingredient of this method is the definition of a family of what we call almost conservation laws. In particular we analyze the Korteweg-de Vries initial value problem and we illustrate in general terms how the ``algorithm’’ that we use to formally generate almost conservation laws can be used to recover the infinitely many conserved integrals that make the KdV an integrable system.