Combinatorics of Sharing Theorems, Stratifications, Bruhat Theory and Shimura Varieties

该项目由组合学计划，刺激竞争研究的既定计划（EPSCoR）以及代数和数论计划共同资助。组合数学是一个数学领域，涉及枚举，理解和表征离散对象中出现的数学结构。 该项目的结果将有助于组合数学和其他数学学科之间日益增长的联系。 这包括扩展经典Coxeter群的共享定理，使用组合方法来推广和理解与志村品种的表示理论有关的身份，并继续PI的成功计划，涉及非交换多项式，这些多项式应用于理解多面体，惠特尼分层空间和完全非负的标志品种。 该奖项的结果有可能深入了解其他科学中的离散结构，包括生物学中DNA测序的非交换性，编码理论中的数学方法以及与机器人运动相关的拓扑表面。 PI一直积极参与研究，教育和推广活动，以促进数学劳动力的增长。 为应对COVID-19大流行造成的隔离，这包括共同组织一个新的区域讲座系列，以重振学生、博士后和教师之间的合作。 PI的研究计划的支持加强了NSF的科学进步，培养新的人才，促进创新和改善社会的目标。更具体地说，该项目包括四个子项目，涉及广泛定义的组合学，加上正在进行的研究生研究项目。 该奖项的结果将有助于组合数学与几何，拓扑，代数和数论之间日益增长的联系，并为组合数学的经典领域做出根本性的贡献。 项目一涉及PI的新工作与埃克斯堡和莫雷尔的扩展共享定理Coxeter安排，并使用赫伯的理论2结构给解剖证明和推广的内在卷。 扩展到其他几何设置将进行研究，包括经常复杂的多面体和Nandakumar-Rao猜想。 项目二涉及PI的联合工作与Ebergborg和Goresky的拓扑面枚举惠特尼分层空间，更一般地说，zeta函数的准分次偏序集。 这拓宽了研究计划，以了解和取得进展的面向量不等式多面体和奇异空间。 PI和Escherborg将经典的cd-指标推广到了满足平衡条件的有向标号图，推广了欧拉分次偏序集的设置，并将Bruhat图族作为特例。 项目III包括一个猜想的平衡有向图，这意味着Billera-Brenti非负猜想的CD指标的Bruhat图。 项目四涉及扩展广义Harish-Chandra特征公式的组合学。 PI将继续她的教育，区域和国家活动，以支持长期目标，吸引，留住和培训更多的数学科学代表性不足的群体，并最终增加劳动力中STEM教育的个人数量。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

This project is jointly funded by the Combinatorics program, the Established Program to Stimulate Competitive Research (EPSCoR), and the Algebra and Number Theory program. Combinatorics is the area of mathematics concerned with enumerating, understanding and characterizing mathematical structures that occur within discrete objects. The results of this project will contribute to the growing connections between combinatorics and other mathematical disciplines. This includes extending sharing theorems of classical Coxeter groups, using combinatorial methods to generalize and understand identities involved with the representation theory of Shimura varieties, and continuing the PI's successful program involving noncommutative polynomials that have applications to understanding polytopes, Whitney stratified spaces and the totally nonnegative flag variety. Results from this award have the potential to give insight into discrete structures in other sciences including the noncommutative nature of DNA sequencing in biology, mathematical methods in coding theory and topological surfaces related to robotic motion. The PI has been vigorously involved in research, educational and outreach activities to foster the growth of the mathematical workforce. As a response to the isolation caused by the COVID-19 pandemic, this includes co-organizing a new regional lecture series to reinvigorate collaboration between students, postdocs and faculty. Support of the PI's research program reinforces the NSF's goals of scientific progress, building new talent, fostering innovation and improving society.More specifically, this project includes four subprojects involving combinatorics broadly defined, plus ongoing graduate research projects. Results from the award will contribute to the growing connections between combinatorics with geometry, topology, algebra and number theory, and to make fundamental contributions to classical areas of combinatorics. Project I involves the PI's new work with Ehrenborg and Morel on extensions of sharing theorems to Coxeter arrangements, and using Herb's theory of 2-structures to give dissection proofs and generalizations to intrinsic volumes. Extensions to other geometric settings will be studied, including regular complex polytopes and the Nandakumar--Rao conjecture. Project II involves the PI's joint work with Ehrenborg and Goresky on topological face enumeration of Whitney stratified spaces, more generally, zeta functions of quasi-graded posets. This widens the research program to understand and make progress on face vector inequalities for polytopes and singular spaces. The PI and Ehrenborg develop a non-homogeneous extension of the classical cd-index to labeled digraphs satisfying a balanced condition to generalize the setting of Eulerian graded posets and include the family of Bruhat graphs as a special case. Project III includes a conjecture for balanced digraphs which implies the Billera--Brenti nonnegativity conjecture for the cd-index of Bruhat graphs. Project IV concerns extending the combinatorics of the generalized Harish-Chandra character formula. The PI will continue her educational, regional and national activities to support the long-range goal to attract, retain and train more under-represented groups in the mathematical sciences and ultimately increase the number of STEM-educated individuals in the workforce.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.