Appears in collection : 2015 - T3 - Mathematical general relativity

The lectures will focus on some of the key issues of mathematical theory of black holes, i. e. rigidity, stability and collapse. The first two issues can be neatly summarized by two well known conjectures. The rigidity conjecture identifies the class of stationary, asymptotically flat solutions to the Einstein vacuum equations (or more general field equations) as the explicit Kerr family. The second asserts that these solutions must be stable under general dynamical perturbations. Though both rigidity and stability are taken for granted in modern astrophysics,they remain far from settled even at a heuristic level, not to speak of rigorous mathematical proofs. The third issue deals with the question of how black holes can form in the first place from regular initial conditions is intimately tied to the concept of trapped surfaces. In the lectures I plan toreview the present state of understanding regarding these problems. This course is part of the Trimester Program “Mathematical general relativity” which took place at the Institut Henri Poincaré in order to celebrate the 100th anniversary of general relativity. See http://philippelefloch. org for further information.