Global well-posedness, scattering and blow-up for the energycritical, focusing, non-linear Schrödinger equation in the radial case

In these lectures I will discuss recent joint works with F. Merle. In them we have developed an approach to the study of non-linear critical problems of dispersive type. The issues studied are global well-posedness and scattering. The approach works for both focusing and defocusing problems, but in these lectures I will concentrate on two focusing problems. The approach proceeds in steps, some of which are general and hence apply to “all problems ” and some which are specific to each particular problem. The concrete problems to be discussed here are the energy critical, focusing non-linear Schrödinger equation and wave equation. I will try to separate both kinds of arguments in the exposition. I will start out…