The mathematical analysis of black holes has been the focus of considerable activity in the past decade from the perspective of the theory of partial differential equations. Much of this work is motivated by the problem of understanding the two celebrated cosmic censorship conjectures in a neighbourhood of the Schwarzschild and Kerr solutions. Recent progress on the behaviour of linear waves on black hole exteriors as well as on the full non-linear vacuum dynamics in the black hole interior puts us at the threshold of a complete understanding of the stabilityand  instabilityproperties of these solutions. This talk will survey some of these developments.

Black Hole: The Mathematical Mystery