Multivariate splines in algebra, analysis, and combinatorics

代数、分析和组合学中的多元样条

• 批准号: 1419103
• 负责人: Amos Ron
• 金额: $ 41万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1419103&HistoricalAwards=false
• 关键词: Multivariate; splines; algebra; analysis; combinatorics

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.

这个研究项目是在数学的多个子学科之间的接口：一方面分析，逼近理论和数据表示;另一方面组合学和代数。该研究涉及到一个新的样条类的建设。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。样条曲线已经成为计算机辅助设计和制造汽车和飞机、打印机排版、自动制图、电影制作以及许多其他领域中不可或缺的工具，通常隐藏在精心制作的软件包的核心中。这个项目将融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。样条函数是一个或多个变量的分段多项式。Zonotopal代数是一种数学方法，它以丰富的代数和分析结构编码组合和几何属性。它目前处理的特殊多面体称为zonotope和它的双超平面安排。在它的核心，人们发现样条理论被称为箱样条，样条定义在zonotopes，可以说是最成功的样条理论在几个变量。Zonotopal代数及其相关的盒样条连接到数学内外的无数主题，包括近似，小波，细分，拟阵，图形，代数几何等等。有证据表明，zonotopal代数应该有一对样条结构：盒样条和另一个样条类的对偶几何。这个项目的目的是恢复这个额外的类，特别是通过扩展zonotopal代数一类的多面体是不变的群体行动。将这种不变性嵌入到zonotopal代数中，并理解这些非交换几何上的正确代数结构和样条构造是本项目的另一个目标。