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Black holes are one of the most celebrated predictions of general relativity. It is widely believed that space-time in the vicinity of these black holes can be described to a suitable approximation by a Kerr metric, a remarkable family of explicit solutions of the Einstein vacuum equations discovered in 1963. Yet even the most basic mathematical questions about the dynamics of the Einstein equations in a neighbourhood of these solutions remain to this day unanswered. Are the Kerr metrics stable as solutions of the Einstein equations? Does gravitational collapse generically lead to black holes, or can so-called "naked singularities" form instead? What happens to observers who enter black hole regions? The latter two questions probe the very limits of the theory and are tied to the more general issue of singularities and the celebrated cosmic censorship conjectures of Penrose.

The last few years has seen intense activity which has brought us to the threshold of a denitive resolution of some of these issues. After intense work by several groups, and using important insights from the physics literature, the decay properties of linear scalar fields on the Kerr exterior backgrounds are now denitively mathematically understood. Concerning the problem of gravitational collapse, a breakthrough recent theorem of Christodoulou proves that trapped surfaces can from from initial data which are arbitrarily dispersed. This work introduces methods for rigorously understanding the Einstein equations in the large field regime, and already has given several applications to other problems involving singularities. Regarding the black hole interior, heuristic, numerical, and now rigorous mathematical theorems have shed light on the singular boundary, denitively showing that it always has a null piece-in contrast to older expectations that singularities should generically be everywhere space-like.