Counting Curves Using the Topology of Moduli Spaces

这位PI将进行代数几何方面的研究，即研究作为多项式方程的解集而产生的空间。这些空间是代数变体，通过检查方程的代数性质和解集的几何来研究它们。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)本身往往是一个代数簇，以这种方式出现的代数簇称为模空间。PI将研究一些特定的模空间，如紧化的泛雅可比空间和稳定映射的Kontsevich模空间，目的是更好地理解它们，并将它们的研究应用于曲线计数等问题。这笔补助金还将支持研究生院的学生和包括南卡罗来纳州数学圈在内的国际学生协会的外展活动。该项目由代数和数论项目和已建立的激励竞争性研究项目(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化的雅可比有了详细的了解，PI将把这种理解应用到高级代数几何中。PI将学习滑轮模空间的算术、几何和拓扑，然后将这些结果应用于解决计数问题(即推进计数几何)。PI将关注的模空间是奇异曲线上的滑轮的模空间或紧化的雅可比，该项目由两个主要部分组成。在第一部分，PI将发展泛紧化雅可比的计数几何，它是稳定曲线上的轮子的模空间，以类似于Grassman变种的Schubert演算的发展。对于第二部分，PI将使用A1同伦理论对曲线进行算术计算。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.