In physical problems like those in geophysics or medical imaging it is often desirable to reconstruct the topology or geometry of a manifold from boundary data. The classical Dirichlet-to-Neumann operator, which is essentially the voltage-to-current map on the boundary of a manifold, has long been known to encode significant geometric information about the manifold. In this talk I will discuss how to recover the cup product on a compact Riemannian manifold with boundary from a generalization of the classical Dirichlet-to-Neumann operator to differential forms.