Non-linear stability of Kerr-de Sitter black holes

I will explain some ideas behind the proof of the global stability of the Kerr-de Sitter family of black holes as solutions of the initial value problem for the Einstein vacuum equations when the cosmological constant is positive. I will describe the general framework which enables us to deal systematically with the diffeomorphism invariance of Einstein's equations, and thus how our solution scheme finds a suitable gauge, within a carefully chosen finite-dimensional family of gauges, in which we can find the global solution. I will also address the issue of finding the mass and the angular momentum of the final blackhole. This talk is based on joint work with András Vasy.