Large Data Dynamics for Nonlinear Dispersive PDEs

We will discuss recent work on wave evolutions for large data. Particular emphasis will be placed on concentration compactness ideas. Amongst others, we will describe a result for wave equations from R^3 minus the unit ball into the sphere S^3 where we can show that any solution approaches the unique harmonic map in its degree class.
Joint work with Cote, Kenig, Lawrie, Nakanishi -- in various combinations.