Counting Curves Using the Topology of Moduli Spaces

使用模空间拓扑计算曲线

• 批准号: 2001565
• 负责人: Jesse Kass
• 金额: $ 18.78万
• 依托单位: University of South Carolina at Columbia
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-05-15 至 2023-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001565&HistoricalAwards=false
• 关键词: Counting; Curves; Using; Topology; Moduli

This PI will conduct research in algebraic geometry which is the study of spaces that arise as solution sets to polynomial equations. These spaces are algebraic varieties, and they are studied both by examining the algebraic properties of the equations and the geometry of the solution sets. An important feature of algebraic geometry is that a collection of algebraic varieties (e.g. the collection of all plane conic curves) often itself is an algebraic variety, and algebraic varieties appearing in this way are called moduli spaces. The PI will study some specific moduli spaces, such as the compactified universal Jacobian and the Kontsevich moduli space of stable maps, with the goal of both better understanding them and applying their study to problems like curve counting. The grant will also support research students and the PI's outreach activities including the South Carolina Math Circle. This project is jointly funded by the Algebra and Number Theory program and the Established Program to Stimulate Competitive Research (EPSCoR). After roughly 60 years of work by many mathematicians, we now have a detailed understanding of how to construct compactified Jacobians, and the PI will apply this understanding to advance algebraic geometry. The PI will study the arithmetic, geometry, and topology of moduli spaces of sheaves and then to apply those results to solve counting problems (i.e. to advance enumerative geometry). The moduli spaces the PI will focus on are moduli spaces of sheaves on singular curves or compactified Jacobians, and the project consists of two broad parts. For the first part, the PI will develop the enumerative geometry of the universal compactified Jacobian, a moduli space of sheaves on stable curves, in a manner analogous to the development of the Schubert calculus of the Grassmannian variety. For the second part, the PI will count curves arithmetically using A1-homotopy theory.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.

该 PI 将进行代数几何研究，即研究作为多项式方程的解集而出现的空间。 这些空间是代数簇，通过检查方程的代数性质和解集的几何形状来研究它们。 代数几何的一个重要特征是，代数簇的集合（例如所有平面二次曲线的集合）往往本身就是一个代数簇，这样出现的代数簇称为模空间。 PI 将研究一些特定的模空间，例如稳定映射的紧致通用雅可比行列式和 Kontsevich 模空间，目的是更好地理解它们并将其研究应用于曲线计数等问题。这笔赠款还将支持研究生和 PI 的外展活动，包括南卡罗来纳州数学圈。该项目由代数和数论项目以及刺激竞争性研究既定项目（EPSCoR）共同资助。经过许多数学家大约 60 年的工作，我们现在对如何构造紧凑雅可比行列式有了详细的了解，PI 将应用这种理解来推进代数几何。 PI 将研究滑轮模空间的算术、几何和拓扑，然后应用这些结果来解决计数问题（即推进枚举几何）。 PI 将关注的模空间是奇异曲线或压缩雅可比行列式上滑轮的模空间，该项目由两个主要部分组成。对于第一部分，PI 将开发通用紧致雅可比行列式的枚举几何，即稳定曲线上滑轮的模空间，其方式类似于格拉斯曼簇的舒伯特微积分的开发。对于第二部分，PI 将使用 A1 同伦理论对曲线进行算术计数。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。