Four lectures on KAM for the non-linear Schrodinger equation

We discuss the KAM-theory for lower-dimensional ton for the non-linear Schrodinger equation with periodic boundary conditions and a convolution potential in dimension d. Central in this theory is the homological equation and a condition on the small divisors often known as the second Melnikov condition. The difficulties related to this condition are substantial when d >= 2. We discuss this difficulty, and we show that a block decomposition and a Toplitz-Lipschitz-property, present for non-linear Schrodinger equation, permit to overcome this difficuly. A detailed proof is given in [EK06].