Black hole non-linear stability: a mathematical ovierview

In this talk, I will survey key geometric and analytic aspects of black hole perturbations, focusing on recent advances in understanding their dynamical stability. Emphasis will be placed on the recent proof of the full nonlinear stability of the slowly rotating Kerr family as solutions to the vacuum Einstein equations—a result obtained in joint work with Klainerman and Szeftel. I will outline the main geometric structures exploited in the proof and highlight the analytical framework that underpins the stability analysis. I will also discuss how the features in the proof change in the presence of a black hole charge.