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

该项目由组合学计划、已建立的激励竞争研究计划(EPSCoR)以及代数和数论计划共同资助。组合学是一个数学领域，涉及列举、理解和描述发生在离散对象中的数学结构。该项目的成果将有助于加强组合学和其他数学学科之间的联系。这包括推广经典Coxeter群的共享定理，使用组合方法推广和理解与Shimura簇的表示理论有关的恒等式，以及继续PI的成功程序，涉及非交换多项式，应用于理解多面体，Whitney分层空间和完全非负的旗簇。这一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测序的非对易性质、编码理论中的数学方法以及与机器人运动相关的拓扑面。国际数学协会一直积极参与研究、教育和外展活动，以促进数学工作人员的增长。作为对新冠肺炎疫情造成的孤立的回应，这包括联合举办一个新的地区性系列讲座，以重振学生、博士后和教师之间的合作。对PI研究计划的支持加强了NSF的科学进步、培养新人才、促进创新和改善社会的目标。更具体地说，该项目包括四个涉及广义组合学的子项目，以及正在进行的研究生研究项目。该奖项的结果将有助于加强组合数学与几何、拓扑、代数和数论之间的联系，并对组合数学的经典领域做出基本贡献。项目I涉及PI与Ehrenborg和Morel在Coxeter排列上共享定理的推广方面的新工作，以及使用Herb的2-结构理论给出对固有体积的解剖证明和推广。我们还将研究对其他几何设置的扩展，包括正则复多面体和Nandakumar-Rao猜想。项目II涉及PI与Ehrenborg和Goresky在Whitney分层空间的拓扑面计数方面的合作，更一般地，关于拟分次偏序集的Zeta函数的工作。这拓宽了研究程序，以了解多面体和奇异空间的面向量不等式并取得进展。PI和Ehrenborg将经典CD-指标的非齐次推广推广到满足平衡条件的标号有向图，从而推广了欧拉分次偏序集的设定，并将Bruhat图族作为特例包括在内。项目III包括一个关于平衡有向图的猜想，这意味着Bruhat图的CD-指标的Billera-Brenti非负性猜想。项目四涉及推广广义Harish-Chandra特征公式的组合学。国际数学奖将继续她的教育、地区和国家活动，以支持在数学科学领域吸引、留住和培训更多代表性不足的群体，并最终增加劳动力中受过STEM教育的个人的长期目标。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.