Algebraic and topological combinatorics

代数和拓扑组合数学

• 批准号: 1002636
• 负责人: Patricia Hersh
• 金额: $ 5.09万
• 依托单位: North Carolina State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-10-15 至 2012-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1002636&HistoricalAwards=false
• 关键词: Algebraic; topological; combinatorics

The PI will study how combinatorial structure of partially ordered sets (posets) gives information about associated algebraic and topological structures. Specifically, she seeks ways to use the combinatorics of a face poset or closure poset together with limited topological information (e.g. information about codimension one cell incidences) to determine topological information such as rational homology or even in some cases homeomorphism type. A main goal is to prove that various stratified spaces whose closure posets are thin and shellable are in fact regular CW complexes. She also aims to construct boundary maps on important classes of posets such as geometric lattices, motivated by potential applications e.g. in theoretical computer science. She will also study representations of simple Lie algebras via an analysis of local structure within the poset that indicates which weight space generators appear with nonzero coefficient upon application of a raising or lowering operator to other weight space generators.In computer science, there is a strong interest both in finding effective algorithms when this is possible, and also in understanding when this is not theoretically possible. It is often difficult to know how to even begin to obtain these impossibility results, but topology combined with combinatorics does have the potential to yield such results, and there have been some striking successes in the past. One of the proposed projects is somewhat in this vein -- to prove impossibility results for the question of which graphs (i.e. which models for networks of computers) admit data sorting algorithms by greedily swapping pieces of neighboring data that are out of order. In another direction, geometric structures such as the Schubert varieties of representation theory/algebraic geometry often have combinatorial data associated to them which is organized into what is called a partially ordered set. Much is known about how properties of the geometric structure force properties on the partially ordered set. The PI seeks a better understanding of how much can be said in the other direction, perhaps allowing some manageable extra data such as information about how edges are attached to 2-dimensional faces, how 2-dimensional faces are attached to 3-dimensional faces, and so on. This could have applications to finding new ways of figuring out what very complicated, high-dimensional geometric objects look like. In addition to continuing to mentor students and run seminars, the PI will also continue organizing conferences designed to foster fruitful new interactions and collaborations between those working in topological combinatorics and in related areas such as combinatorial representation theory and combinatorial commutative algebra.

PI 将研究偏序集（偏序集）的组合结构如何提供有关相关代数和拓扑结构的信息。 具体来说，她寻求使用面偏序集或闭合偏序集的组合与有限的拓扑信息（例如，关于余维单细胞发生率的信息）的方法来确定拓扑信息，例如理性同源性，甚至在某些情况下的同胚类型。 主要目标是证明闭偏集很薄且可壳化的各种分层空间实际上是规则的 CW 复合体。 她还致力于在重要的偏序集（例如几何格子）类别上构建边界图，其动机是潜在的应用，例如：在理论计算机科学中。 她还将通过对偏序集中的局部结构进行分析来研究简单李代数的表示，该结构表明在将提升或降低运算符应用于其他权重空间生成器时，哪些权重空间生成器会出现非零系数。在计算机科学中，人们对寻找可能的有效算法以及理解这在理论上不可能的情况都抱有浓厚的兴趣。 通常很难知道如何开始获得这些不可能的结果，但拓扑与组合学相结合确实有可能产生这样的结果，并且过去已经取得了一些惊人的成功。 拟议的项目之一在某种程度上就是这样的——通过贪婪地交换无序的相邻数据片段来证明哪些图（即计算机网络的哪些模型）承认数据排序算法的问题的不可能性结果。 在另一个方向上，诸如舒伯特形式的表示论/代数几何之类的几何结构通常具有与之相关的组合数据，这些数据被组织成所谓的部分有序集。 关于几何结构的性质如何影响偏序集的性质，人们已经了解很多。 PI 寻求更好地理解在另一个方向上可以说多少内容，也许允许一些可管理的额外数据，例如有关边缘如何附加到 2 维面部、2 维面部如何附加到 3 维面部的信息，等等。 这可以用于寻找新的方法来弄清楚非常复杂的高维几何对象的样子。 除了继续指导学生和举办研讨会之外，PI 还将继续组织会议，旨在促进拓扑组合学以及组合表示理论和组合交换代数等相关领域的人员之间富有成效的新互动和合作。