The Jacobian, the square root and the set ∞

In these notes, we wish to present a new approach to the problem of prescribing Jacobian determinants in dimension two: we restrict ourselves to the case the datum is a finite sum of Dirac masses. The main point is to show that we may relate this problem to the search of harmonic maps into a singular space shaped as the symbol ∞. The later problem in turn is closely linked to questions in complex analysis. A large part of the paper is devoted to a presentation of these mathematical objects and their connections.