Bruhat and balanced graphs, manifolds, partitions and affine permutations

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。本提案由代数、几何和拓扑组合五个广泛的研究课题组成。第一个项目是将Coxeter系统的强Bruhat图的结果扩展到更一般的平衡图集。PI最近将Billera和Brenti关于Bruhat图的cd索引的工作扩展到平衡图，并表明Kazhdan-Lusztig多项式，一个重要的拓扑和表示理论不变量，也扩展到这种设置。第二个项目是利用cd-index对流形正则细分的面入射数据进行研究。PI将研究凸多面体经典结果的流形类比，包括确定cd指数系数的不等式，以Zaslavsky关于超平面排列的开创性工作的精神发展规则流形排列的理论，并将Stanley的球壳性概念扩展到流形。受Wachs在d-可分格上的工作的启发，第三个项目是从偏置同调、壳性和表示理论的角度来研究限制分格。第四个项目是第二类斯特林数和贝尔数的道林类似物的经典枚举研究。第五个项目是研究仿射排列组的组合排列统计，如超差、下降和主要指数。组合学本质上是一个跨学科的研究领域，将数学和科学的许多领域联系在一起。这一建议进一步扩展了组合学的范围。例如，Kazhdan-Lusztig多项式是最初在拓扑学中定义的深度不变量。在这个项目中，我们将从组合的角度来分析它们，以进一步加深我们对它们的理解。适用于球体形状的多面体的理论将扩展到具有更复杂拓扑结构的其他流形。发展我们的拓扑学观点很重要，因为现代物理学的很大一部分都集中在研究空间的拓扑学上。加深我们对排列统计和基本组合枚举的理解可以帮助我们分析和识别大量基因组数据中的模式，以及改善包括互联网在内的通信。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This proposal is composed of five broad research topics in algebraic, geometric and topological combinatorics. The first project is to expand results about the strong Bruhat graph of Coxeter systems to the more general setting of balanced graphs. The PI has recently extended Billera and Brenti's work on the cd-index of Bruhat graphs to balanced graphs and showed the Kazhdan-Lusztig polynomials, an important topological and representation theoretic invariant, also extend to this setting. The second project is a study of the face incidence data of regular subdivisions of manifolds via the cd-index. The PI will investigate manifold analogues of classical results for convex polytopes, including to determine inequalities for the coefficients of the cd-index, to develop a theory of regular manifold arrangements in the spirit of Zaslavsky's seminal work on hyperplane arrangements, and to extend Stanley's notion of spherical shellability to manifolds. Motivated by Wachs' work on the d-divisible partition lattice, the third project is to examine the restricted partition lattice from the standpoint of poset homology, shellability and representation theory.The fourth project is a classical enumerative study of Dowling analogues of the Stirling numbers of the second kind and the Bell numbers. The fifth project is to investigate combinatorial permutation statistics, such as excedances, descents and the major index, for the group of affine permutations.Combinatorics is inherently an interdisciplinary field of study linking many areas of mathematics and the sciences. This proposal further expands the range of combinatorics. For example, the Kazhdan-Lusztig polynomials are a deep invariant originally defined in topology. In this project they will be analyzed from a combinatorial perspective to further enhance our understanding of them. Theories that apply to polytopes, which are sphere-like objects, will be extended to other manifolds which have more complicated topological structure. Developing our topological perspective is important since a large part of modern-day physics is focused on studying the topology of space. Deepening our understanding of permutation statistics and basic combinatorial enumeration may help us to analyze and recognize patterns in vast genome data, as well as to improve communications, including the internet.