EPSCoR: Combinatorics of Sharing Theorems, Stratifications, Bruhat Theory and Shimura Varieties (CFDA # 47.083)

NSF Proposal: Combinatorics of Sharing Theorems, Stratifications, Bruhat Theory and Shimura Varieites Margaret A. Readdy Project Summary Overview. This proposal consists of four research projects involving combinatorics broadly de?ned, plus on- going graduate research projects. The aim is to strengthen connections to geometry, topology, al- gebra and number theory, and 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 are proposed, 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 strati?ed spaces, more generally, zeta functions of quasi-graded posets. This widens the research program to under- stand 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 satis- fying 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. Intellectual Merit. Results from the grant will contribute to the growing connections between combinatorics and other mathematical disciplines. This includes extending sharing theorems of classical Coxeter groups, us- ing 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 strati?ed spaces and the totally nonnegative ?ag variety. Results from this grant 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 robotics. Broader Impacts. The PI is vigorously involved in activities to foster the growth of the mathematical workforce via the PI’s research, educational and outreach. These reinforce the NSF’s goals of scienti?c progress, building new talent, fostering innovation and improving society. The PI has graduated PhD students from Appalachia, as well as female students, and actively mentors a female person of color who is now in her third year of graduate school in Applied Mathematics at Virginia Tech. The PI gives virtual and in-person plenary lectures in regional, national and international venues. These include the Triangle Lectures in Combinatorics in the North Carolina research triangle, a three-lecture series at a midwest regional combinatorics conference for graduate students and the 2019 EUROCOMB in Slovakia. The PI has also led two research groups at BIRS for female combinatorialists, and has assisted with the mathematical and mentoring portions of the Institute for Advanced Study Women and Mathematics (WAM) program for ?ve years. The PI organizes in-person and virtual outreach in local schools and a large scale state math festival in 2023. The PI has served as a Guest Editor for the March 2018 Women’s History Month issue of the Notices, and co-wrote a chapter on the IAS Women and Mathematics Program in the AMS/MAA Classroom Resource Material “Count Me In: Community and Belonging in Mathematics”, all with an eye to connect with the public. Support of this proposal will enable the PI to continue and expand these activities.