The AdS instability conjecture is a conjecture related to the initial-boundary value problem for the vacuum Einstein equations with a negative cosmological constant. It states that generic, arbitrarily small perturbations to the initial data of the AdS spacetime, under evolution by the vacuum Einstein equations with reflecting boundary conditions on conformal infinity, lead to the formation of black holes. This conjecture was introduced in 2006 by Dafermos and Holzegel and, since then, it has attracted a vast amount of numerical and heuristic works by several authors, starting from the work of Bizon and Rostworowski in 2011. These works have been focused mainly on the simpler setting of the spherically symmetric Einstein-scalar field system.

In this talk, we will provide the first rigorous proof of the AdS instability conjecture in the simplest possible setting, namely for the spherically symmetric Einstein-massless Vlasov system, in the case when the Vlasov field is moreover supported only on radial geodesics. This system is equivalent to the Einstein--null dust system, allowing for both ingoing and outgoing dust. In order to overcome the "trivial" break down occuring once the null dust reaches the centre $r=0$, we will study the evolution of the system in the exterior of an inner mirror with positive radius $r_{0}$ and prove the conjecture in this setting. After presenting our proof, we will briefly explain how the main ideas can be extended to more general matter fields.