  In this paper, we consider the global well-posedness of the defocusing,
$L^{2}$ - critical nonlinear Schr{\"o}dinger equation in dimensions $n \geq 3$.
Using the I-method, we show the problem is globally well-posed in $n = 3$ when
$s > {2/5}$, and when $n \geq 4$, for $s > \frac{n - 2}{n}$. We combine energy
increments for the I-method, interaction Morawetz estimates, and almost
Morawetz estimates to prove the result.
