LINEAR STABILITY OF BLACK HOLES [d'apres M. Dafermos et I. Rodnianski]

Nonlinear stability of black holes is one of the central open problem in General Relativity today. Heuristic, physics type, arguments have been advanced to establish the first necessary step, i.e. linear stability; yet none of them were either convincing or sufficiently robust to apply to the nonlinear setting. This situation has changed dramatically in the last few years through new geometric methods introduced by a number of authors. In my talk I will try to present the main ideas behind these results and focus, in particular, on the remarkable new results of M. Dafermos and I. Rodnianski on the boundedness and decay of solutions to the wave equation in a Kerr background.