From Topology to Combinatorics and Back

从拓扑到组合数学并返回

• 批准号: 0900912
• 负责人: Edward Swartz
• 金额: $ 15万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2012-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0900912&HistoricalAwards=false
• 关键词: Topology; Combinatorics; Back

ABSTRACTPrincipal Investigator: Swartz, Edward B. Proposal Number: DMS - 0900912 Institution: Cornell UniversityTitle: From Topology to Combinatorics and BackThe primary research aim of this project is to study the interplay between topology, geometry and combinatorics. Four areas will receive special emphasis: triangulations of compact manifolds and closely related spaces, Cohen-Macaulay complexes, finite linear quotients of spheres, and matroids. In the last century a tremendous amount of research has been directed toward unlocking the topology and geometry of manifolds. However, questions of a combinatorial character have remained largely unanswered. What is the minimum number of facets required to triangulate a given manifold? What methods are there to construct manifolds with particular combinatorial properties? Given limits on the combinatorics of a triangulation, what does that imply about the possible geometries on the manifold? Similar questions, where even less is known, apply to spaces with singularities which include algebraic varieties, quotients of group actions, or limits of Riemannian manifolds. Cohen-Macaulay complexes and posets are one of the fundamental structures of algebraic combinatorics with applications ranging from partitions to network reliability. Their enumerative properties were worked out in the 1970's by the ground breaking work of Hochster, Reisner and Stanley. What happens if further constraints are put on these spaces? As previously shown by the PI, complexes with more structure, such as finite buildings, geometric lattices and matroid complexes, have strong restrictions on their f-vectors. How common are these additional structures and constraints? Finite linear quotients of spheres lie at the intersection of representation theory, topology, geometry and combinatorics. How are the topology and geometry of the quotient space related to the combinatorics and algebra of the representation theoretic data? How does one study complicated spaces and structures? One approach is to approximate them with simpler objects. For instance, an n-dimensional space might be represented as a collection of n-simplices - the higher dimensional analogues of triangles and tetrahedrons. How complicated are these representations? How many of the smaller pieces are needed? How is this reflected in the shape and geometry of the original object? Are there computationally practical ways of producing these models? These are types of questions addressed in this research.

摘要项目负责人：Swartz， Edward B.项目编号：DMS - 0900912机构：康奈尔大学题目：从拓拓学到组合学再返回本项目的主要研究目的是研究拓拓学、几何学和组合学之间的相互作用。将特别强调四个领域：紧流形和紧密相关空间的三角剖分、Cohen-Macaulay复合体、球体的有限线性商和拟阵。在上个世纪，大量的研究都是为了解开流形的拓扑和几何。然而，组合性质的问题在很大程度上仍未得到解答。对给定流形进行三角剖分所需的最小面数是多少？有什么方法可以构造具有特定组合性质的流形？给定三角形组合的极限，这对流形上可能的几何形状意味着什么？类似的问题，在更少的地方，适用于奇异的空间，包括代数变量，群作用的商，或黎曼流形的极限。Cohen-Macaulay复合体和偏序集是代数组合学的基本结构之一，其应用范围从分区到网络可靠性。它们的枚举性质是在20世纪70年代由Hochster， Reisner和Stanley的开创性工作得出的。如果对这些空间施加进一步的限制会发生什么？正如前面的PI所示，具有更多结构的配合物，如有限建筑物，几何晶格和矩阵配合物，对其f向量有很强的限制。这些额外的结构和约束有多普遍？球的有限线性商是表示论、拓扑学、几何学和组合学的交叉点。商空间的拓扑和几何与表示理论数据的组合学和代数有什么关系？如何研究复杂的空间和结构？一种方法是用更简单的对象来近似它们。例如，一个n维空间可能被表示为n个简单体的集合——即三角形和四面体的高维类似物。这些表示有多复杂？需要多少个小块？这是如何反映在形状和几何的原始对象？有没有计算上实用的方法来产生这些模型？这些都是本研究要解决的问题类型。