RUI: Combinatorial Algebraic Geometry: Curves and Their Moduli

RUI：组合代数几何：曲线及其模

• 批准号: 2101861
• 负责人: Linda Chen
• 金额: $ 20.71万
• 依托单位: Swarthmore College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101861&HistoricalAwards=false
• 关键词: RUI; Combinatorial; Algebraic; Geometry; Curves

Algebraic geometry is a central area of mathematics that studies varieties, which are geometric objects defined by systems of polynomial equations. Moduli theory aims to understand specific varieties by considering how they behave in a family of such varieties. In particular, a moduli space consists of all geometric objects of a particular type. This research project consists of problems that arise from the fruitful interactions between algebraic geometry and new developments in other fields of mathematics such as combinatorics, which is concerned with organizing and analyzing discrete structures. This project will fund undergraduate research and the PI will continue efforts towards broadening participation of members of underrepresented groups in the mathematical sciences. The projects aim to better understand moduli spaces that are combinatorially rich in nature and their cohomology theories. Fundamental objects of investigation include degeneracy loci, homogeneous spaces such as Grassmannians and flag varieties, the affine Grassmannian, and the moduli space of curves. More specifically, projects include the study of degeneracy loci and their motivic classes, with applications to Brill-Noether theory; quantum cohomology and quantum K-theory of homogeneous spaces and connections to the affine Grassmannian; base point free classes on the moduli space of curves arising from Gromov-Witten theory and from representation theory; and other problems in algebraic geometry and algebraic combinatorics, including several for undergraduate students.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.

代数几何是研究变量的数学中心领域，变量是由多项式方程系统定义的几何对象。模理论的目的是通过考虑它们在这样的变种族中的行为来理解特定的变种。特别地，模空间由所有特定类型的几何对象组成。该研究项目包括代数几何与其他数学领域（如组合学）的新发展之间富有成效的相互作用所产生的问题，组合学涉及组织和分析离散结构。该项目将资助本科生的研究，而PI将继续努力扩大代表性不足群体成员在数学科学领域的参与。这些项目旨在更好地理解组合丰富的模空间及其上同调理论。研究的基本对象包括简并轨迹、齐次空间（如Grassmannian和flag变体）、仿射Grassmannian以及曲线的模空间。更具体地说，项目包括退化轨迹及其动机类的研究，并应用于Brill-Noether理论；齐次空间的量子上同调和量子k理论及其与仿射格拉斯曼的联系由Gromov-Witten理论和表示理论导出曲线模空间上的无基点类以及代数几何和代数组合中的其他问题，其中包括几个面向本科生的问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Motivic classes of degeneracy loci and pointed Brill‐Noether varieties

简并位点和尖头布里尔-诺特变体的动机类别

• DOI: 10.1112/jlms.12547
• 发表时间: 2022
• 期刊: Journal of the London Mathematical Society
• 作者: Anderson, Dave;Chen, Linda;Tarasca, Nicola
• 通讯作者: Tarasca, Nicola

On an Equivalence of Divisors on $\overline {\text {M}}_{0,n}$ from Gromov-Witten Theory and Conformal Blocks

关于来自 Gromov-Witten 理论和共形块的 $overline { ext {M}}_{0,n}$ 上的除数等价

• DOI: 10.1007/s00031-022-09752-6
• 发表时间: 2022
• 期刊: Transformation Groups
• 影响因子: 0.7
• 作者: Chen, L.;Gibney, A.;Heller, L.;Kalashnikov, E.;Larson, H.;Xu, W.
• 通讯作者: Xu, W.