A recent article in the QuantaMagazine (link) discusses a paper of Georgios Moschidis (arXiv:1812.04268) who proved instability of Anti-de Sitter space-time for a certain Einstein-matter system.
Recall that the Anti-de Sitter space-time is the maximally symmetric solution of the vacuum Einstein equations in the presence of a negative cosmological constant. One can attach a boundary-at-infinity to Anti-de Sitter space-time and then study initial-boundary value problems for different kinds of boundary conditions. Due to the well-posedness of this initial-boundary value problem one can also study the dynamics of solutions. In 2006 Dafermos and Holzegel formulated an instability conjecture for vacuum Einstein equations with a reflecting boundary condition, which attracted a lot of attention in the last years.
The first rigorous proof of the instability conjecture was achieved by Georgios Moschidis in an earlier paper (arXiv:1704.08681). The novelty of the new paper is that he now constructs an example of an unstable Anti-de Sitter space-time without a so-called inner mirror at a finite radius in the space-time.
If you are not an expert in that field of mathematics, I recommend reading the article in the QuantaMagazine!