Concentration Compactness for the Critical Maxwell-Klein-Gordon Equation

The Maxwell-Klein-Gordon equation models the interaction of an electromagnetic field with a charged particle field. We discuss a proof of global regularity, scattering and a priori bounds for solutions to the energy critical Maxwell-Klein-Gordon equation relative to the Coulomb gauge for essentially arbitrary smooth data of finite energy. The proof is based upon a novel "twisted" Bahouri-Gérard type profile decomposition and a concentration compactness/rigidity argument by Kenig-Merle, following the method developed by Krieger-Schlag in the context of critical wave maps. This is joint work with Joachim Krieger.