The focusing CCM model is a completely integrable PDE that describes a continuum limit of a particle gas interacting pairwise through an inverse square potential.  This system is well-posed in the scaling-critical space L^2 below the mass of the soliton, but above this threshold there are solutions that blow up in finite time.

In this talk, we will discuss some new and existing results about solutions below the soliton mass threshold, and outline some of the main ideas in their proofs.  This is based on joint work with Rowan Killip and Monica Visan.