  We derive rigorously, for both R^2 and [-L, L]^2, the cubic nonlinear
Schrodinger equation in a suitable scaling limit from the two-dimensional
many-body Bose systems with short-scale repulsive pair interactions. We first
prove convergence of the solution of the BBGKY hierarchy, corresponding to the
many-body systems, to a solution of the infinite Gross-Pitaevskii hierarchy,
corresponding to the cubic NLS; and then we prove uniqueness for the infinite
hierarchy, which requires number-theoretical techniques in the periodic case.
