ÃÛÌÒTV

In the first part of the talk I will review the black hole stability problem with emphasis on how geometric features of black hole geometries enter into the mathematical analysis. This part will focus mostly on the linear scalar wave equation. In the second part, I will discuss a recent proof (obtained in collaboration with Dafermos and Rodnianski) of the linear stability of the Schwarzschild solution under gravitational perturbations. This involves a linearization of the Einstein equation in a double null gauge, and exploiting a hierarchy of gauge invariant quantities that decouple from the system of equations.