Research in Algebraic Combinatorics

PI继续她的代数和拓扑方面的研究与部分有序集（偏集）和单调图的性质相关的单纯复合体。偏序集拓扑理论在组合学和其他数学分支（如拓扑、代数和几何）之间提供了深刻而基本的联系。这个项目有五个部分。前三部分与研究交换代数中的一种新的偏序集运算的拓扑性质有关，称为Rees积。通过研究两个非常简单的序集的Rees积，PI和John Shareshian发现了一些非凡的枚举恒等式和代数恒等式。在列举恒等式中，最引人注目的是对众所周知的欧拉多项式的主要指数和超越指数的联合分布的q-类比的推测。在第4部分中，PI提出了一个众所周知的划分格的同调与不连通图的复合体的同调之间的关系的k-类比。PI和Shareshian对它应该是什么有一个精确的推测，涉及到所谓的1 mod k划分偏置和非k边连通图的复合体。在第五部分，PI建议继续她对匹配复合体、棋盘复合体和变异的研究。这些复合体出现在不同的设置，如群论，离散几何和交换代数。代数组合是数学的一个领域，它试图在组合学和涉及代数的纯数学领域之间建立联系。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、排列和分析离散构型的科学。通信网络是称为图的基本离散配置的一个例子。图和其他离散构型出现在数学、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。

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.