An emerging theory of Zonotopal Algebra is a framework for studying various algebraic, combinatorial, and analytic objects associated to a linear map from a higher-dimensional space to a lower-dimensional one. It provides connections between polynomial structures such as partition functions and positive structures such as f- and h-vectors of matroids. This perspective gives formulas for volumes and lattice point enumerators of zonotopes, hence the name.
This framework was inspired by the theory of Box Splines, piecewise polynomial functions whose chambers are determined by such a linear map. There are wall-crossing formulas that explain how box splines change when moving from one chamber to another. Bos splines can be thought of as fiber volume functions, as they measure the volume of the preimage of their argument under the linear map in question, when the preimage is restricted to the unit cube.
Zonotopal algebra thus connects analytic, algebraic and matroidal structures generated by linear maps. It examines a family of polynomial ideals, spaces of polynomials annihilated by these ideals, geometry of the associated hyperplane arrangements, tilings of the associated zonotope, and many related phenomena from a unified point of view.
Information on the NSF DMS grant Zonotopal Algebra and Combinatorics at UC Berkeley.