In the presence of confinement, the Einstein field equations are expected to exhibit turbulent dynamics. One way to introduce confinement to the equations is by imposing a negative value for the cosmological constant. In this setting, the AdS instability conjecture claims the existence of arbitrarily small perturbations to the initial data of Anti-de Sitter spacetime which, under evolution by the vacuum Einstein equations with reflecting boundary conditions at conformal infinity, lead to the formation of black holes after sufficiently long time.

In this talk, I will discuss a number of problems in connection with the turbulent dynamics of the equations in the asymptotically AdS setting. I will also present a rigorous proof of the AdS instability conjecture in the setting of the spherically symmetric Einstein-scalar field system.