The goal of the work reported upon in this
talk is to understand the nonlinear Schr\"odinger (NLS) blowup dynamics for rough initial data. Much of the known theory for NLS blowup relies upon energy conservation and is thus restricted to $H^1$ solutions, even though the evolution problem is well-posed in $L^2$. Two new results in the $L^2$-critical on $\R^2$ setting
will be described: 1. For all $s$ satisfying $s_Q < s < 1$, $H^s$ blowup solutions concentrate at least the mass of the ground state
soliton into a point at blowup time. 2. $L^2$ blowup solutions concentrate at least a fixed amount (conjectured to be the mass of the ground state) of $L^2$ mass at blowup time.
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