  We prove two finite dimensional approximation results and a symplectic
non-squeezing property for the Korteweg-de Vries (KdV) flow on the circle T.
The nonsqueezing result relies on the aforementioned approximations and the
finite-dimensional nonsqueezing theorem of Gromov. Unlike the work of Kuksin
which initiated the investigation of non-squeezing results for infinite
dimensional Hamiltonian systems, the nonsqueezing argument here does not
construct a capacity directly. In this way our results are similar to those
obtained for the NLS flow by Bourgain. A major difficulty here though is the
lack of any sort of smoothing estimate which would allow us to easily
approximate the infinite dimensional KdV flow by a finite-dimensional
Hamiltonian flow. To resolve this problem we invert the Miura transform and
work on the level of the modified KdV (mKdV) equation, for which smoothing
estimates can be established.
