Geometric asymptotics and beyond

The analysis of Einstein's field equations in the context of Penrose's notion of asymptotic simplicity, which was originally introduced to provide a geometric setting for the investigation of gravitational radiation, reveals the existence of conformal representations of the field equations which imply evolution equations that are hyperbolic up to and beyond conformal infinity. This peculiar feature of the equations allows us to formulate various well-posed initial and initial-boundary value problems for the conformal field equations which lead to general large scale existence and strong stability results as well as to sharp results on the asymptotic behaviour and the conformal extensibility of solutions to Einstein's field equations. We discuss the physical relevance of these results and various open questions.