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

University Of Kentucky Research Foundation, Lexington KY

Investigators

Abstract

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.