Topology and combinatorics of arrangements

Louisiana State University, Baton Rouge LA

Investigators

Abstract

The theory of arrangements has connections to various fields of mathematics and beyond, from algebraic and geometric topology, algebraic geometry, and combinatorics to applied topology, physics, and engineering. This award supports research on nonlinear arrangements and on elucidating the topological implications of the associated combinatorial data. The mentoring, organizational, and outreach activities of the PI will impact the education of undergraduate and graduate students and support the participation and advancement of women and other underrepresented groups in the STEM fields. The project is jointly funded by Topology and the Established Program to Stimulate Competitive Research (EPSCoR). A central theme in arrangement theory is the interplay between the topology of the complement to a family of submanifolds in a given space and the combinatorics of its intersection data. While there are considerable developments in the case of an arrangement of hyperplanes in a vector space, there has been a recent surge of activity on nonlinear arrangements. This project explores new directions and develops new techniques in the study of arrangements, especially arrangements of hypersurfaces in a complex torus or a product of elliptic curves. This involves using ideas motivated by topology and algebraic geometry to generalize tools from matroid theory, and then applying this combinatorial framework to study topological invariants of arrangement complements and configuration spaces. 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.