Much of my research has concerned the stability problem for black holes:
Do initially small perturbations of Kerr black holes evolve under the Einstein vacuum equations to a nearby member of the Kerr family?
Even the linear aspects of this problem (linearised gravity on a fixed Schwarzschild or Kerr background) are not yet understood, despite intense study since the late 1950s.
On the other hand, we do now have a definitive understanding at least of the linear scalar aspects of the problem (the scalar wave equation on a fixed background as above), as a result of contributions from many researchers over the past few years.
The mathematics surrounding the latter problem is already exceedingly rich and is intimately tied to the characteristic physical/geometric features associated with black holes: the celebrated redshift at the horizon, the phenomenon of superradiance, and trapped null geodesics circling the black hole, near which energy can concentrate for long times.
For an elementrary introduction to the problem, with many references, one can consult my lecture notes, written jointly with Igor Rodnianski:
Lectures on black holes and linear waves,
which accompanied a course given at the Zürich Clay Summer School of Summer 2008.
Some more recent developments in this subject are described below: