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I will discuss recent progress in understanding gravitational signatures of the discreteness of the energy spectrum of a toy quantum black hole. The black hole is described by the Sachdev-Ye-Kitaev (SYK) model, a quantum mechanical system displaying quantum chaos. The SYK model's low energy limit is given, via gauge/gravity duality, by Jackiw-Teitelboim gravity in 2 spacetime dimensions. This gravitational theory computes Weil-Petersson volumes of the moduli space of Riemann surfaces. Mirzakhani's recursion for these volumes has been related to the Eynard-Orantin topological recursion for a certain matrix integral. The eigenvalues of this matrix are the energies of the black hole, which therefore have random matrix statistics, characteristic of a quantum chaotic system with discrete spectrum. The properties of the eigenvalues are probed by a spacetime analog of a D-brane. Analysis of another type of D-brane yields new large genus asymptotic formulas for Weil-Petersson volumes. This is joint work with Phil Saad and Douglas Stanford.