Research in Algebraic Combinatorics

Wachs dms -0302310奖摘要：PI继续研究与部分有序集（偏集）和单调图性质相关的简单复的代数和拓扑方面。偏序集拓扑理论起源于1964年罗塔关于偏序集的莫比乌斯函数的著名论文，它为组合学和其他数学分支（如拓扑学、代数和几何）之间提供了深刻而基本的联系。单调图性质的拓扑研究的意义首先在Kahn， Saks和Sturtevant 1984年对算法复杂性理论中回避猜想的素幂情况的证明中得到证明。这个项目有三个部分。在第一部分中，PI探索了三个迷人的组合复合体之间的联系，这些复合体出现在各种背景下的文献中；即非完美匹配复形，1 mod k划分偏置和Whitehouse树复形的推广。在第二部分中，PI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背景下，如群论、离散几何和交换代数。在第三部分，PI继续她的工作，在一些有趣的猜想汉隆处理李代数同调。预计这三个部分的研究将涉及到拓扑和代数组合新技术的发展。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。这些跨学科的联系有助于丰富和推进组合学和其他领域。组合学是计算、排列和分析离散构型的科学。通信网络和系统发育树是称为图的基本离散配置的例子。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大的作用。

Abstract for award of Wachs DMS-0302310The PI continues her investigation of algebraic and topological aspects of simplicial complexes associated with partially ordered sets (posets) and monotone graph properties. The theory of poset topology, which grew out of the famous 1964 paper of Rota on the Mobius function of a partially ordered set, provides a deep and fundamental link between combinatorics and other branches of mathematics such as topology, algebra and geometry. The significance of the topological study of monotone graph properties was first demonstrated in Kahn, Saks, and Sturtevant's 1984 proof of the prime power case of the evasiveness conjecture in algorithmic complexity theory. There are three parts to this project. In the first part, the PI explores connections between three fascinating combinatorial complexes, which have appeared in the literature in various contexts; namely the no-perfect matching complex, the 1 mod k partition poset and a generalization of the Whitehouse tree complex. In the second part, the PI continues her study of the matching complex, the chessboard complex and variations. These complexes arose in diverse settings such as group theory, discrete geometry and commutative algebra. In the third part, the PI continues her work on some intriguing conjectures of Hanlon dealing with Lie algebra homology. It is expected that the research in all three parts will involve the development of new techniques in topological and algebraic combinatorics.Algebraic combinatorics is an area of mathematics that seeks to establish connections between combinatorics and fields of pure mathematics that involve algebra. These interdisciplinary connections serve to enrich and advance combinatorics and the other fields. Combinatorics is the science of counting, arranging and analyzing discrete configurations. Communications networks and phylogenic trees are examples 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 expanding role in these fields.