Multivariate splines in algebra, analysis, and combinatorics

本研究项目是数学多个分支学科的交叉点：一方面是分析、近似理论和数据表示；另一方面是组合学和代数。该研究涉及到一个新的样条类的构造。过去对样条函数的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要的贡献。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，以及在许多其他领域，样条已经成为不可或缺的工具，通常隐藏在精心制作的软件包的核心。该项目将融合来自不同数学领域的知识和技能，以提供新的多元样条构造以及代数和几何方面的新理论结果。样条函数是一个或几个变量的分段多项式。分区代数是一种数学方法，它在丰富的代数和解析结构中编码组合和几何性质。它目前处理的特殊多面体称为带拓扑及其对偶超平面排列。在它的核心，我们发现样条理论被称为盒样条，样条是在共面上定义的，可以说是几个变量中最成功的样条理论。分区代数及其相关的盒样条与数学内外的无数主题有关，包括近似、小波、细分、拟阵、图、代数几何等。有证据表明，分区代数应该有一对样条构造：盒样条和对偶几何上的另一类样条。本项目旨在恢复这一额外的类别，特别是通过将带拓扑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共格代数中，并在这些非交换几何上理解正确的代数结构和样条构造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.