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In this paper we prove that the focusing, $d$-dimensional mass criticalnonlinear Schr{\"o}dinger initial value problem is globally well-posed andscattering for $u_{0} \in L^{2}(\mathbf{R}^{d})$, $\| u_{0}\|_{L^{2}(\mathbf{R}^{d})} < \| Q \|_{L^{2}(\mathbf{R}^{d})}$, where $Q$ is theground state, and $d \geq 1$. We first establish an interaction Morawetzestimate that is positive definite when $\| u_{0} \|_{L^{2}(\mathbf{R}^{d})} <\| Q \|_{L^{2}(\mathbf{R}^{d})}$, and has the appropriate scaling. Next, wewill prove a frequency localized interaction Morawetz estimate similar to theestimates made in \cite{D2}, \cite{D3}, \cite{D4}. See also \cite{CKSTT4} forthe energy critical case. Since we are considering an $L^{2}$ - criticalinitial value problem we will localize to low frequencies.

Global well-posedness and scattering for the mass critical nonlinear Schr{\"o}dinger equation with mass below the mass of the ground state