Counting Curves Using the Topology of Moduli Spaces

这个PI将进行代数几何的研究，这是研究空间出现的解决方案集多项式方程。 这些空间是代数簇，它们都是通过检查方程的代数性质和解集的几何来研究的。 代数几何的一个重要特征是代数簇的集合（例如所有平面圆锥曲线的集合）本身往往是一个代数簇，以这种方式出现的代数簇被称为模空间。 PI将研究一些特定的模空间，例如紧致化的泛雅可比矩阵和稳定映射的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。该赠款还将支持研究学生和PI的推广活动，包括南卡罗来纳州数学圈。该项目由代数和数论计划以及刺激竞争研究的既定计划（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化雅可比行列式有了详细的了解，PI将把这种理解应用于推进代数几何。PI将研究层的模空间的算术、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）。PI将关注的模空间是奇异曲线或紧化雅可比行列式上的层的模空间，该项目由两个主要部分组成。对于第一部分，PI将开发通用紧化雅可比矩阵的枚举几何，稳定曲线上的层的模空间，以类似于格拉斯曼簇的舒伯特微积分的开发方式。对于第二部分，PI将使用A1同伦理论计算曲线。该奖项反映了NSF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。

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.