Topology and combinatorics of arrangements

从代数和几何拓扑学、代数几何学、组合学到应用拓扑学、物理学和工程学，排列理论与数学的各个领域都有联系。该奖项支持对非线性排列和阐明相关组合数据的拓扑含义的研究。非政府组织的指导、组织和外联活动将影响本科生和研究生的教育，并支持妇女和其他在STEM领域任职人数不足的群体的参与和提高。该项目由拓扑学和已建立的激励竞争研究计划(EPSCoR)共同资助。排列理论的一个中心主题是给定空间中一族子流形的补集的拓扑与其交集数据的组合之间的相互作用。虽然在向量空间中超平面的排列方面有了相当大的发展，但最近关于非线性排列的活动激增。这个项目探索了排列研究的新方向和新技术，特别是复杂环面或椭圆曲线乘积中的超曲面的排列。这涉及到使用拓扑学和代数几何的思想来推广拟阵理论的工具，然后应用这个组合框架来研究排列互补和配置空间的拓扑不变量。这一奖项反映了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。

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.