Kerr black holes are solutions of Einstein's theory of general relativity and are theoretically predicted to describe astrophysical stationary black holes. A mathematical proof of their nonlinear stability remains a fundamental open problem in the subject. I will present a new geometric framework to address the stability of the Kerr solution to gravitational perturbations in the full sub-extremal range of black hole parameters. The main features of the framework will be illustrated in the context of the linearised theory, which serves as a fundamental building block in nonlinear applications. The talk includes joint work with R. Teixeira da Costa (University of Cambridge).
