  Eighth-BPS local operators in N=4 SYM are dual to quantum states arising from
the quantization of a moduli space of giant gravitons in AdS5xS5. Earlier
results on the quantization of this moduli space give a Hilbert space of
multiple harmonic oscillators in 3 dimensions. We use these results, along with
techniques from fuzzy geometry, to develop a map between quantum states and
brane geometries. In particular there is a map between the oscillator states
and points in a discretization of the base space in the toric fibration of the
moduli space. We obtain a geometrical decomposition of the space of BPS states
with labels consisting of U(3) representations along with U(N) Young diagrams
and associated group theoretic multiplicities. Factorization properties in the
counting of BPS states lead to predictions for BPS world-volume excitations of
specific brane geometries. Some of our results suggest an intriguing
complementarity between localisation in the moduli space of branes and
localisation in space-time.
