This book chapter is published open access.
We survey some recent mathematical progress in understanding singularities arising in solutions to the Einstein equations. After some quick discussions of background material, we focus on the following three topics:
- constructions of singular solutions to the Einstein vacuum equations,
- the singularity structure in the interior of generic dynamical black holes and the relation to the strong cosmic censorship conjecture,
- the formation of trapped surfaces, instabilities for the Einstein vacuum equations, and the relation to singularities.