# Mathematical relativity and differential geometry

Differential geometry and geometric analysis provide the mathematical underpinnings of GR and also occur in many other fields of physics. While GR is formulated in differential geometric language, the resulting field equations are geometric partial differential equations. Existence, uniqueness and stability questions for such equations are therefore of crucial importance in relativity. A good understanding of the analytic properties of these equations is also indispensable for the construction of algebraic quantum field theories on curved space-times.

GR predicts that very dense matter configurations bend space-time so much that light is trapped. The resulting objects, known as black holes, play an essential role in astrophysics. Most galaxies contain a super-massive black hole, and black hole mergers are one of the most important sources of gravitational radiation. The Kerr black hole solution is expected to be the unique black hole model in GR and is therefore of vast importance in astrophysics. In order to establish the relevance of the Kerr model, it is essential to prove that it is dynamically stable. This leads to the black hole stability problem that is very challenging and complex, and an active research topic around the world with the AEI playing a key role in it.

Self-gravitating matter systems in GR describe a wide range of phenomena including compact stars, galaxies, as well as matter in a cosmological context. A combination of numerical and analytical techniques is necessary to investigate for instance the formation and stability of steady states as well as the asymptotic behaviour of complex systems. There has been a great deal of progress recently in the analysis of non-linear evolution equations modelling these phenomena, and it is now important to apply these to problems motivated by gravitational physics.