Multivariate splines in algebra, analysis, and combinatorics

这项研究项目是在数学的多个子学科之间的交界处：一方面是分析、逼近理论和数据表示；另一方面是组合学和代数。这项研究涉及到一个新的样条族的构造。过去对样条函数理论和实践的数学研究导致了数学界对科学和技术的一些最重大的贡献。在汽车和飞机的计算机辅助设计和制造中，在打印机排版的生产中，在自动制图中，在电影制作中，在许多其他领域，样条线已经成为不可或缺的工具，这些工具往往隐藏在复杂的软件包的核心中。这个项目将结合不同数学领域的知识和技能，提供新的多元样条构造以及代数和几何的新理论结果。样条函数是一种或多种变量的分段多项式。Zonotopal代数是一种数学方法，它将组合和几何性质编码为丰富的代数和分析结构。它目前处理被称为带状多面体的特殊多面体及其双超平面排列。在它的核心，人们发现了被称为盒样条的样条线理论，定义在纬向上的样条线，可以说是在几个变量中最成功的样条线理论。Zonotopal代数及其相关的盒样条连接到数学内外的无数主题，包括逼近、小波、细分、拟阵、图、代数几何等等。有证据表明，Zonotopal代数应该有一对Spline结构：盒Spline和对偶几何上的另一类Spline。这个项目的目的是恢复这个额外的类，特别是通过将zonotopal代数扩展到一类在群作用下不变的多面体。将这种不变性嵌入到zonotopal代数中，并在这些非交换几何上理解正确的代数结构和样条结构是本项目的另一个目标。

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.