From Topology to Combinatorics and Back

从拓扑到组合数学并返回

• 批准号: 0600502
• 负责人: Edward Swartz
• 金额: $ 11.85万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0600502&HistoricalAwards=false
• 关键词: Topology; Combinatorics; Back

The primary theme of this project is to study the interplay between topology, geometry and combinatorics. Four areas will receive special emphasis: triangulations of compact manifolds, Cohen-Macaulay complexes, finite linear quotients of spheres, and matroids. It has been over 100 years since the introduction of the Euler-Poincare formula, yet the combinatorial properties of triangulations of compact manifolds remain largely unknown. Indeed, a complete characterization of all possible f-vectors (which record the number of faces in each dimension) is not known for any manifold of dimension five or more. The possible f-vectors of Cohen-Macaulay simplicial complexes have been known since the foundational work of Hochster, Reisner and Stanley in the 1970's. However, the classification of face counts for important subclasses, such as spheres and doubly Cohen-Macaulay complexes, remains a fundamental open problem in algebraic and topological combinatorics. Even though the representation theory of finite groups is an extremely well developed subject, tools for computing many topological invariants, such as Betti numbers, for the corresponding spherical quotients are practically nonexistent. Matroids are a combinatorial abstraction of linear independence which can be modeled through a variety of topological spaces. Their enumerative properties have many applications including coloring and flows on graphs, network reliability, and the topology of hyperplane arrangementsTwo recurrent and complementary ideas in mathematics are the approximation of continuous phenomena using discrete data, and modeling the latter with smooth objects, such as spheres or polyhedra. The guiding principle of this project is that the combination of these two techniques is a powerful method of approaching both situations.For instance, compact manifolds, which are frequently presented as solutions to nonsingular polynomial equations in several variables, can also be examined by using simplicial complexes which are easier to encode into a computer. In turn, this allows a deeper study of the shape and geometry of the original space.

这个项目的主要主题是研究拓扑学，几何学和组合学之间的相互作用。 四个领域将得到特别强调：三角形的紧凑型流形，科恩-麦考利复合物，有限线性的领域，拟阵。自从欧拉-庞加莱公式被引入到现在已经有100多年了，但是紧致流形的三角剖分的组合性质仍然是未知的。 事实上，所有可能的f向量（记录每个维度上的面数）的完整特征对于任何五维或更高维的流形都是未知的。 Cohen-Macaulay单纯复形的可能的f-向量自Hochster、Reisner和Stanley在20世纪70年代的基础工作以来就已经被人们所知。 然而，分类的面计数的重要子类，如领域和双科恩-麦考利复合物，仍然是一个基本的开放问题，在代数和拓扑组合。 尽管有限群的表示论是一个非常发达的学科，但用于计算许多拓扑不变量的工具，如Betti数，对于相应的球面不变量实际上是不存在的。 拟阵是线性独立性的组合抽象，可以通过各种拓扑空间建模。 它们的枚举属性具有 许多应用包括图的着色和流动、网络可靠性和超平面的拓扑结构。数学中两个经常出现的互补思想是使用离散数据近似连续现象，以及使用光滑对象（如球体或多面体）对后者建模。 这个项目的指导原则是，这两种技术的结合是一个强大的方法来接近这两种情况。例如，紧凑的流形，这是经常提出的非奇异多项式方程的解决方案，在几个变量，也可以检查使用单纯复形，这是更容易编码到计算机。 反过来，这允许对原始空间的形状和几何形状进行更深入的研究。