  In recent years, the AdS instability conjecture, put forward by
Dafermos-Holzegel in 2006, has attracted a substantial amount of numerical and
heuristic studies. Following the pioneering work of Bizon-Rostworowski,
research efforts have been mainly focused on the study of the spherically
symmetric Einstein-scalar field system. The first rigorous proof of the
instability of AdS in the simplest spherically symmetric setting, namely for
the Einstein-null dust system, was obtained in [G. Moschidis, A proof of the
instability of AdS for the Einstein-null dust system with an inner mirror,
2017, arXiv:1704.08681]. In that work, the evolution was restricted to the
exterior of an inner mirror placed at $r=r_{0}>0$, in order to circumvent the
trivial break down of the Einstein-null dust system occurring at $r=0$; in that
setting, additional considerations necessitated $r_{0}$ to shrink to $0$ with
the size of the initial perturbation.
  In this paper, we establish the instability of AdS for the Einstein-massless
Vlasov system in spherical symmetry. This will be the first proof of the AdS
instability conjecture for an Einstein--matter system which is well-posed for
regular initial data in the standard sense, without the addition of an inner
mirror. New difficulties associated with the Einstein-massless Vlasov system
(such as the need for control on the paths of non-radial geodesics in a large
curvature regime) force us to develop a strategy of proof which is
fundamentally different from the one employed in the case of the Einstein--null
dust system, albeit still based on the interaction of beams of matter in
physical space.
