Around 50 years ago, physicists Belinski, Khalatnikov and Lifshitz provided heuristics describing the behaviour of near-singularity solutions to the Einstein equations as spacelike, local and oscillatory. Vaguely speaking, they suggest that dynamics corresponding to different spatial points on the singularity decouple and resemble a chaotic cascade of so-called “Kasner epochs”. In this talk, we present a mathematical formulation of this “BKL Conjecture” and progress towards the conjecture in the case of Gowdy symmetric spacetimes.

Solutions to general relativity with a negative cosmological constant have received significant attention due to the conjectured AdS/CFT correspondence, a particularly well-understood example of which is exhibited in 2+1 dimensions. I will review known vacuum solutions to general relativity with a negative cosmological constant in 2+1 dimensions and discuss the difficulties in defining mass, which are resolved via minimisation using a positive energy theorem. I will present a gluing theorem for vacuum time-symmetric general-relativistic initial data sets in two spatial dimensions. By gluing two given time-symmetric vacuum initial data sets at conformal infinity, we obtain new time-symmetric vacuum initial data sets. I will sketch the derivation of the mass formulae of the resulting manifolds. Our gluing theorem yields complete manifolds with any mass aspect function, which are smooth except for one conical singularity. Based on joint work with P. T. Chruściel.