Generalized witness sets The fundamental data structure in numerical algebraic geometry is that of a witness set, which is considered to be an instantiation of Weil's notion of a generic point. Reinterpreting a witness set in terms of duality of the intersection pairing in intersection theory leads to a generalization of the notion that makes sense on many spaces and leads to a general notion of a witness set. This notion is most productive for homogenous spaces, such as projective spaces, Grassmannians, and their products. After exlaining the general theory, I will sketch what this means for the Grassmannian.