This research project is at the interface among multiple sub-disciplines of mathematics: analysis, approximation theory, and data representation on the one hand; combinatorics and algebra on the other hand. The research involves the construction of a new spline class. Past mathematical research on the theory and practice of spline functions led to some of the most significant contributions of the mathematical community to science and technology. Splines have become indispensable tools in computer-aided design and manufacturing of cars and airplanes, in the production of printers' typesets, in automated cartography, in the production of movies, and in many other areas, often concealed at the core of elaborate software packages. This project will merge knowledge and skills from disparate areas of mathematics to provide new multivariate spline constructions as well as new theoretical results in algebra and geometry.
Spline functions are piecewise-polynomials in one or several variables. Zonotopal algebra is a mathematical methodology that encodes combinatorial and geometric properties in rich algebraic and analytic structures. It presently handles the special polytope known as a zonotope and its dual hyperplane arrangement. At its core one finds the spline theory known as box splines, splines that are defined over zonotopes, arguably the most successful spline theory in several variables. Zonotopal algebra and its associated box splines are connected to a myriad of topics inside and outside mathematics, including approximation, wavelets, subdivision, matroids, graphs, algebraic geometry and more. There is evidence that zonotopal algebra should have a pair of spline constructs: box splines and another spline class over the dual geometry. This project aims to recover this additional class, in particular through extending zonotopal algebra to a class of polytopes that are invariant under group actions. Embedding such invariance into zonotopal algebra and understanding the correct algebraic structures and spline constructions over these non-commutative geometries is another goal of this project.
