Elliott Lieb Princeton University It is well known that any given density ρ(x) can be realized by a finite energy determinantal wave function for N particles. The question addressed here is whether any given density ρ(x) and current density j(x) can be simultaneously realized by a (finite kinetic energy) determinantal wave function. In case the velocity field v(x)=j(x)/ρ(x) is curl free, we provide a solution for all N, and we provide an explicit upper bound for the energy. If the velocity field is not curl free, there is a finite energy solution for all N≥4, which uses the recent smooth extension of the Hobby-Rice theorem, but we do not provide an explicit energy bound in this case. For N=2 we provide an example of a non curl free velocity field for which there is a solution, and an example for which there is no solution. The case N=3 with a non curl free velocity field is left open. (Joint work with Robert Schrader.)