From Topology to Combinatorics and Back

主要研究人员：爱德华·斯沃茨提案编号：DMS-0900912机构：康奈尔大学标题：从拓扑学到组合学和背景本项目的主要研究目的是研究拓扑学、几何和组合学之间的相互作用。四个领域将得到特别强调：紧致流形和紧密相关空间的三角剖分，Cohen-Macaulay复形，球面的有限线性商，以及拟阵。在上个世纪，大量的研究被指向解锁流形的拓扑和几何。然而，组合性质的问题在很大程度上仍然没有得到回答。对给定的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殊组合性质的流形？给出三角剖分的组合学限制，这意味着流形上可能的几何图形是什么？类似的问题，在更少人知道的情况下，适用于具有奇点的空间，包括代数簇、群作用商或黎曼流形的极限。Cohen-Macaulay复形和偏序集是代数组合学的基本结构之一，其应用范围从划分到网络可靠性。它们的计数性质是由Hochster，Reisner和Stanley在20世纪70年代的S的开创性工作中得出的。如果对这些空间施加进一步的限制，会发生什么？如PI图所示，具有更多结构的复形，如有限建筑物、几何格子和拟阵复形，对它们的f-向量有很强的限制。这些额外的结构和约束有多常见？球面的有限线性商是表示论、拓扑学、几何学和组合学的交集。商空间的拓扑和几何与表示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物体来近似它们。例如，n维空间可以表示为n维单纯的集合--三角形和四面体的高维类似物。这些陈述有多复杂？需要多少较小的碎片？这是如何反映在原始对象的形状和几何体中的？有没有计算上可行的方法来产生这些模型？这些都是这项研究中涉及的问题类型。

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.