Combinatorics, Algebra and Topology of simplicial complexes

单纯复形的组合学、代数和拓扑

• 批准号: 0500748
• 负责人: Isabella Novik
• 金额: $ 9.85万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-15 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500748&HistoricalAwards=false
• 关键词: Combinatorics; Algebra; Topology; simplicial; complexes

This proposal describes four projects aimed to deepenour understanding of the combinatorics, algebra and topology ofsimplicial complexes through the study of their size (or face numbers).This includes work on the Upper Bound Theorem that has its origins inthe theory of convex polytopes, and the study of the face numbers andBetti numbers of balanced simplicial complexes, flag complexes, andcomplexes with symmetry.Combinatorics is an active field of mathematics today that has closeconnections with many other subjects such asalgebra, geometry and topology, optimization, statistics,etc. inside mathematics, and computer science, high-energy physics, andbiology outside. Continuing exploring and developing new cross-pollinations and interactions between combinatorics and other areas ofmathematics is a central intellectual pursuit of this project.Specifically, this proposal concerns simplicial complexes --- finiteobjects that provide the easiest way to represent multi-dimensional shapesin a computer through their triangulations --- and various bounds on thesizes of such complexes. The minimal size of a triangulation has taken onpractical significance, since it directly affects the complexity ofcomputations involving a particular shape.

这项建议描述了四个项目，旨在通过研究单纯形复形的大小(或面数)来加深对它们的组合学、代数和拓扑学的理解。这包括起源于凸多面体理论的上限定理的工作，以及平衡单纯形复形、旗形复形和对称复形的面数和Betti数的研究。组合学是当今数学的一个活跃领域，它与数学内部的许多其他学科，如代数、几何和拓扑、最优化、统计学等，以及计算机科学、高能物理和生物学有着密切的联系。继续探索和发展组合学和其他数学领域之间的新的交叉授粉和相互作用是这个项目的核心智力追求。具体地说，这个建议涉及单纯复形-有限对象，通过它们的三角剖分提供在计算机中表示多维形状的最简单方式-以及对这样的复形的大小的各种界限。三角剖分的最小尺寸具有实际意义，因为它直接影响涉及特定形状的计算的复杂性。