This project is devoted to studies of zonotopal algebra and its combinatorial applications. The methodology of zonotopal algebra is already known to play a role in Approximation Theory (box splines), Commutative Algebra (fat point ideals, Cox rings), Enumerative Combinatorics (matroids and simplicial complexes), Convex Geometry (zonotopes and hyperplane arrangements), Graph Theory (Tutte polynomials, parking functions), and Algebraic Geometry (hyperkaehler toric varieties, de Concini-Procesi wonderful models). This project seeks to advance the theory of zonotopal algebra in the following three ways: (1) develop a discrete counterpart to the existing continuous theory, (2) apply zonotopal algebra to problems of algebraic and enumerative combinatorics, (3) build the infrastructure for larger-scale collaboration on the subject.
This research effort should advance an important area at the crossroads of pure and applied mathematics. Results of this research may also have applications to the mathematics of large networks and traffic flow, including wireless communication, problems of statistical mechanics, scheduling, logistics and discrete optimization. The project will involve graduate students and postdoctoral scholars in and outside UC Berkeley. Junior participants will be encouraged to interact with each another within a research seminar at UC Berkeley as well as with more senior experts in the field during mutual collaboration visits. This should contribute to the integration of research training and education.
