Black Hole Stability Problems in GR

In this talk, I will give an overview of the stability problems for black hole solutions, starting with the mode stability results in black hole perturbation theory in the 80’s to more recent mathematical proofs, as the linear and the fully non-linear stability of black hole solutions require new mathematical techniques. Finally, I will present some aspects of our recent proof with Klainerman and Szeftel of the non-linear stability of the slowly rotating Kerr black hole.