the supergravity limit of AdS/CFT. We are interested in solving for the bulk dynamics where will give ourselves the freedom to specify boundary conditions on $\Sigma_D$. The second boundary condition (which is necessary to zero-in onto a unique solution) will be specified by demanding regularity in the interior of the spacetime. We have schematically depicted the set-up in . In the large $N$ limit, the specification of the problem thus is tantamount to solving classical partial differential equations (PDEs) in AdS with a Dirichlet boundary condition imposed on various fields at the hypersurface $\Sigma_D$. The question we would like to know the answer to is simply: “What is the problem that we are solving in the CFT language?”

![Schematic representation of the Dirichlet problem we consider in this paper. The Dirichlet surface is taken to be at some value $r = r_D$ where we impose boundary conditions on the fields. The solutions will further be constrained by requiring that they be regular on any putative horizon ${\mathcal H}^+$ (shown in the figure) or the origin. The question we are after is what is the boundary image of this Dirichlet data?[]{data-label="f:setup"}](Dir-schema)

(0,0) (-4.8,1.36)[$\Sigma_D$]{} (-8.4,2.3)[${\mathcal H}^+$]{} (-5.24,-0,36)[data = $\hat{\mathfrak X}$]{} (-1.7,-0.36)[data = 