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The AdS instability conjecture is a conjecture about the
initial-boundary value problem in general relativity. It states that
there exist arbitrarily small perturbations to the initial data of AdS
spacetime which, under evolution by the vacuum Einstein equations with
reflecting boundary conditions on conformal infinity, lead to the
formation of black holes after sufficiently long time. In recent
years, a vast amount of numerical and heuristic works have been
dedicated to the study of the instability of AdS, focusing mainly on
the simpler setting of the spherically symmetric Einstein--scalar
field system.
In this talk, I will present a rigorous proof of the AdS
instability conjecture in the setting of the spherically symmetric
Einstein--massless Vlasov system. The construction of the unstable
family of initial data will require working in a low regularity
setting, carefully designing a family of initial configurations of
localised Vlasov beams and estimating the exchange of energy taking
place between interacting beams over long period of times. If time
permits, I will briefly discuss how the main ideas of the proof can be
extended to more general matter fields.