Global well-posedness and scattering conjecture for the defocusing mass-critical nonlinear Schrödinger equation

Consider the defocusing, mass-critical nonlinear Schrödinger initial value problem

with initial data $u_0\in L^2(\mathbf{R}^d)$, where $d\geq 1$. A solution scatters forward or backward in time if it approaches a free solution $e^{it\Delta}u_+$ or $e^{it\Delta}u_-$, respectively, in $L^2(\mathbf{R}^d)$ as $t\to\infty$ or $t\to-\infty$. **Global well-posedness and scattering conjecture.** For $d\geq 1$, the defocusing, mass-critical nonlinear Schrödinger initial value problem

is globally well-posed for $u_0\in L^2(\mathbf{R}^d)$ and all solutions scatter to a free solution as $t\to\pm\infty$.

Small initial data are known to yield global well-posedness and scattering in the two-dimensional case, while the conjecture asserts this for arbitrary $L^2$ initial data and every dimension. The paper addresses the two-dimensional case.