Inverse optimization problems: designing potential functions for target structures

I will describe the inverse optimization problem for potential energy of configurations on the sphere. Namely, given a finite set of points C on the sphere (e.g. the vertices of a cube), can we specify a potential function for which C is a global minimum among all the configurations of |C| points on the sphere? I wil describe necessary and sufficient conditions, a heuristic algorithm to find the potential function in a fixed space of functions, and also give examples of some familiar configurations for which we can find natural potentials which are provably minimized at these codes. This is joint work with Henry Cohn.