The quantum vortices of a superfluid are described as nodal lines of a solution to the time-dependent Gross-Pitaevskii equation. Experiments and extensive numerical computations show that quantum vortices cross, each of them breaking into two parts and exchanging part of itself for part of the other. This phenomenon is known as quantum vortex reconnection, and usually leads to a change of topology of the quantum vortices. In this talk I will show that, given any initial and final congurations of quantum vortices (which do not need to be topologically equivalent) and any conceivable way of reconnecting them (that is, of transforming one into the other), there is a Schwartz initial datum whose associated solution is smooth and realizes this specific vortex reconnection scenario. Key for the proof of this result is a new global approximation theorem for the linear Schrodinger equation, and the construction of pseudo-Seifert surfaces in spacetime. This is based on joint work with Alberto Enciso.