  We prove symplectic non-squeezing (in the sense of Gromov) for the cubic
nonlinear Schr\"odinger equation on $\R^2$. This is the first symplectic
non-squeezing result for a Hamiltonian PDE in infinite volume. As the
underlying symplectic Hilbert space is $L^2(\R^2)$, this requires working with
initial data in this space. This space also happens to be scaling-critical for
this equation. Thus, we also obtain the first unconditional symplectic
non-squeezing result in such a critical setting.
  More generally, we show that solutions of this PDE can be approximated by a
finite-dimensional Hamiltonian system, despite the wealth of non-compact
symmetries: scaling, translation, and Galilei boosts. This approximation result
holds uniformly on bounded sets of initial data. Complementing this
approximation result, we show that all solutions of the finite-dimensional
Hamiltonian system can be approximated by the full PDE.
  A key ingredient in these proofs is the development of a general methodology
for obtaining uniform global space-time bounds for suitable Fourier truncations
of dispersive PDE models.
