Spacetime singularity: on the BKL Proposal outside homogeneity

We know from the Penrose-Hawking singularity theorems that singularities are a robust prediction of the (classical) Einstein equations. In a series of works, Belinski, Khalatnikov and Lifshitz (often abbreviated to BKL) provide heuristics suggesting that such a singularity is generically (a) spacelike and silent, meaning different spatial points don’t communicate in the limit, and (b) oscillatory and governed by a chaotic cascade of so-called Kasner epochs and BKL bounces at each spatial point. In this talk, we review these heuristics and propose a mathematical interpretation of these results, together with some recent progress in certain symmetry classes.