I will describe recent evidence that anti-de Sitter spacetime is
nonlinearly unstable. Generic small but finite perturbations eventually
form small black holes. There are special solutions, called geons, which
remain nonsingular and do not collapse. I will also describe (weaker)
evidence that black holes in anti-de Sitter spacetime are also unstable.
The endstate may involve a violation of cosmic censorship. It would be
of great interest to check all of these claims using numerical
relativity.
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