Stability and instability of asymptotically anti-de Sitter black holes

I will introduce the problem of black hole stability in the context of a negative cosmological constant, describing some of the geometric and analytic challenges characteristic of this problem. In the second part, I will discuss recent work with Olivier Graf (Grenoble) establishing linear stability of Schwarzschild-anti de Sitter (AdS) black holes to gravitational perturbations. This is the statement that solutions to the linearisation of the Einstein equations $\text{Ric}=-\frac{3}{{\ell }^2}g$ around a Schwarzschild-AdS metric arising from regular initial data and with standard Dirichlet boundary conditions imposed at the conformal boundary (inherited from fixing the conformal class of the non-linear metric) remain globally uniformly bounded on the black hole exterior and in fact decay inverse logarithmically to a linearised Kerr-AdS metric.