  In this paper, we propose a new approach to study the BPS dynamics in N=4
supersymmetric U(N) Yang-Mills theory on R X S^3, in order to better understand
the emergence of gravity in the gauge theory. Our approach is based on
supersymmetric, space-filling Q-balls with R-charge, which we call R-balls. The
usual collective coordinate method for non-topological scalar solitons is
applied to quantize the half and quarter BPS R-balls. In each case, a different
quantization method is also applied to confirm the results from the collective
coordinate quantization. For finite N, the half BPS R-balls with a U(1)
R-charge have a moduli space which, upon quantization, results in the states of
a quantum Hall droplet with filling factor one. These states are known to
correspond to the ``sources'' in the Lin-Lunin-Maldacena geometries in IIB
supergravity. For large N, we find a new class of quarter BPS R-balls with a
non-commutativity parameter. Quantization on the moduli space of such R-balls
gives rise to a non-commutative Chern-Simons matrix mechanics, which is known
to describe a fractional quantum Hall system. In view of AdS/CFT holography,
this demonstrates a profound connection of emergent quantum gravity with
non-commutative geometry, of which the quantum Hall effect is a special case.
