Topology and combinatorics of arrangements

排列理论与数学的各个领域以及其他领域都有联系，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。该奖项支持非线性安排和阐明相关组合数据的拓扑含义的研究。PI的指导，组织和推广活动将影响本科生和研究生的教育，并支持妇女和其他代表性不足的群体在STEM领域的参与和进步。该项目由拓扑学和刺激竞争研究的既定计划（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拓扑与其相交数据的组合学之间的相互作用。虽然在向量空间中的超平面的安排的情况下有相当大的发展，但最近非线性安排的活动激增。该项目探索了新的方向，并开发了新的技术，在研究安排，特别是安排超曲面在一个复杂的环面或产品的椭圆曲线。这涉及到利用拓扑和代数几何的思想，从拟阵理论中推广工具，然后应用这种组合框架来研究排列补数和配置空间的拓扑不变量。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值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.