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 2. Estimating blow up speed. What can we say in general about the speed of concentration when a singularity forms? Very little is known in general. I will give a detailed proof of blow up of the critical norm which will allow us to introduce the concept of Liouville rigidity theorem.