The Gross--Pitaevskii equation is a nonlinear Schrödinger equation that models the behavior of a Bose-Einstein condensate. The quantum vortices of the condensate are defined by the zero set of the wave function at time $t$. In this talk we will present recent work about how these quantum vortices can break and reconnect in arbitrarily complicated ways. As observed in the physics literature, the distance between the vortices near the breakdown time, say $t=0$, scales like the square root of $t$: it is the so-called $t^{1/2}$ law. At the heart of the proof lies a remarkable global approximation property for the linear Schrödinger equation. The talk is based on joint work with Daniel Peralta-Salas.