Can you hear the shape of dual geometries?

We compute the sub-leading terms in the Tian-Yau-Zelditch asymptotic expansion of the partition function for dual giant gravitons on AdS5 $\times$ L5 and provide a bulk interpretation in terms of curvature invariants. We accomplish this by relating the partition function of dual giant gravitons to the Hilbert series for mesonic operators in the CFT. The coefficients of the subleading terms encode integrated curvature invariants of L5. In the same spirit of Martelli, Sparks and Yau, we are able to compute these integrated curvature invariants without explicit knowledge of the Sasaki-Einstein metric on L5. These curvature invariants contribute to the 1/N^2 corrections of the difference of the 4D anomaly coefficients a and c recently found by Liu and Minasian, which we now have a purely field theoretic method of calculating.