Title: The algebra of box splines, hyperplane arrangements, and zonotopes

Date: 04/13/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Olga Holtz, UC Berkeley

Zonotopal algebra is a framework for studying various algebraic, combinatorial, and analytic objects associated to a linear map $\phi: \R^N \rightarrow \R^n$, where $n\le N$. This unified perspective gives formulas for volumes and lattice point enumerators of certain zonotopes, among other things.
This framework was inspired by the theory of box splines, which are piecewise-polynomial functions supported on zonotopes, whose chambers are determined by the matrix $X$ of the map $\phi$. Box splines can be thought of as fiber volume functions, as they measure the volume of the $(N-n)$-dimensional preimage of their $n$-dimensional argument, where the preimage is restricted
to the ``box' $[0,1]^N$.
I'll explain how the theory of zonotopal algebra connects these analytic phenomena to the algebraic properties of the linear map $X$. In particular, how the matroidal structure of $X$ is related to:
1. a family of polynomial ideals associated to $X$,
2. the kernels of those ideals, i.e., the spaces of polynomials annihilated by those ideals,
3. the discrete geometry of the associated hyperplane arrangement, and
4. the tilings of the associated zonotope.
This new line of research allows to study combinatorial and algebraic objects using techniques of analysis. Examples include recent results of de Concini, Procesi, Vergne, Moci, Lenz, and others.