The description of singularity formation in non linear evolution equations has attracted a considerable amount of interest in the last fifty years, with a tremendous acceleration since the turn of the Millennium. This series of lectures is intended as an introduction to the subject. First two lectures will present some elementary graduate level materials on non linear dispersive equations, last two will be more advanced with the presentation of state of art results and open problems in the field.

Lecture 1: The (NLS) model. The non linear Schrodinger equation is a canonical model of non linear dispersive equation. We will review standard results of local existence, global existence and scattering in the defocusing case, and virial blow up in the focusing regime. We will in particular introduce the concept of critical/super critical problem.