Research in Algebraic Combinatorics

代数组合学研究

• 批准号: 0604562
• 负责人: Michelle Wachs
• 金额: --
• 依托单位: University of Miami
• 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: 2006-06-01 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604562&HistoricalAwards=false
• 关键词: Research; Algebraic; Combinatorics

The PI continues her investigation of algebraic and topological aspects of simplicialcomplexes associated with partially ordered sets (posets) and monotone graph properties. The theory of poset topology provides a deep and fundamental link between combinatorics and other branches of mathematics such as topology, algebra and geometry. There are five parts to the project. The first three parts are connected with the study of topological properties of a new poset operation coming from commutative algebra, called Rees product. By studying the Rees product of two very simple posets, the PI and John Shareshian have discovered some remarkable enumerative and algebraic identities. The most striking of the enumerative identities is a conjectured q-analog of a well-known identity for the Eulerian polynomials in terms of the joint distribution of the major index and the excedance index. In Part 4, the PI proposes to obtain a k-analog of a well-known relationship between the homology of the partition lattice and the homology of the complex of graphs that are not connected. The PI and Shareshian have a precise conjecture on what that should be, involving the so called1 mod k partition poset and the complex of graphs that are not k-edge connected. In Part 5, the PI proposes to continue her study of the matching complex, the chessboard complex and variations. These complexes arise in diverse settings such as group theory, discrete geometry and commutative algebra. Algebraic combinatorics is an area of mathematics that seeks to establish connections between combinatorics and fields of pure mathematics that involve algebra. The idea is to use these connections to enrich combinatorics and the other fields. Combinatorics is the science of counting, arranging and analyzing discrete configurations. A communications network is an example of a fundamental discrete configuration called a graph. Graphs and other discrete configurations arise in various fields of mathematics, computer science, physics and biology. Combinatorial methods are playing an increasing role in these fields.

PI继续她的调查代数和拓扑方面的simplicialcomplex与偏序集（偏序集）和单调图形属性。偏序集拓扑理论在组合数学与其他数学分支如拓扑学、代数学和几何学之间提供了一个深刻而基础的联系。 该项目有五个部分。前三部分是关于交换代数中一种新的偏序集运算--Rees积的拓扑性质的研究。 通过研究两个非常简单的偏序集的Rees积，PI和John Shareshian发现了一些显着的枚举恒等式和代数恒等式。 最引人注目的枚举恒等式是欧拉多项式的一个著名恒等式在主指数和超额指数的联合分布方面的一个简化的q-模拟。 在第4部分中，PI提出了一个k-模拟的一个众所周知的关系之间的同源性的分区格和同源性的复杂的图是不连接的。 PI和Shareshian有一个精确的猜想，应该是什么，涉及所谓的1模k分割偏序集和复杂的图，不是k-边连接。 在第5部分，PI建议继续研究匹配复合体，棋盘复合体和变体。这些复合物出现在不同的设置，如群论，离散几何和交换代数。代数组合学是一个数学领域，旨在建立组合学和涉及代数的纯数学领域之间的联系。 我们的想法是利用这些联系来丰富组合学和其他领域。 组合学是计算、排列和分析离散结构的科学。 通信网络就是一个被称为图的基本离散配置的例子。 图和其他离散配置出现在数学、计算机科学、物理学和生物学的各个领域。 组合方法在这些领域中发挥着越来越重要的作用。