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将在代数几何形状上进行研究，该研究是针对多项式方程的解决方案的空间的研究。 这些空间是代数品种，通过检查方程的代数特性和溶液集的几何形状来研究它们。 代数几何形状的一个重要特征是，代数品种（例如，所有平面圆锥曲线的集合）通常是代数品种，并且以这种方式出现的代数品种称为模量空间。 PI将研究一些特定的模量空间，例如稳定地图的压缩通用雅各布和肯特维奇模量空间，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。该赠款还将支持研究学生和PI的外展活动，包括南卡罗来纳州数学圈。该项目由代数和数字理论计划和启发竞争性研究的既定计划共同资助（EPSCOR）。经过许多数学家的大约60年的工作，我们现在对如何构建压缩的雅各布人有了详细的了解，PI将应用这种理解来推进代数的几何形状。 PI将研究滑轮模量空间的算术，几何和拓扑，然后将这些结果应用于解决计数问题（即推进枚举几何形状）。 PI将重点关注的模量空间是在奇异曲线或压缩的Jacobians上的带束带的模量空间，该项目由两个宽部分组成。在第一部分中，PI将以稳定曲线上的带束线束的模束空间发展，以类似于格拉斯曼尼亚品种的舒伯特演算的方式。在第二部分中，PI将使用A1-HOMOTOPY理论对算术进行计数曲线。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响审查标准通过评估来支持的。