  In this work we investigate the unique properties of ultracold Bose gases in
one and two dimensions. In two dimensions, we present simulations of the
Berezinskii-Kosterlitz-Thouless (BKT) phase transition using the projected
Gross-Pitaevskii equation formalism. We confirm the presence of the BKT phase
by observing algebraic decay in the first-order correlation function, unbinding
of vortex pairs, and behaviour of the superfluid fraction. In one dimension we
use a transverse Gaussian ansatz and the Lagrangian approach to derive an
effective equation of motion for the one dimensional Bose gas at zero
temperature.
