Double Ramification Cycles and Tautological Classes

双分支循环和同义反复类

• 批准号: 2301506
• 负责人: Aaron Pixton
• 金额: $ 19万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2301506&HistoricalAwards=false
• 关键词: Double; Ramification; Cycles; Tautological; Classes

Algebraic geometry is a branch of mathematics that studies geometric properties of spaces defined by polynomial equations. One of the key objects in algebraic geometry is the moduli space of curves, a geometric space whose points correspond to types of algebraic curves. This moduli space has many connections to other areas of mathematics and to physics (where the algebraic curves are the strings appearing in string theory). One way to study the moduli space of curves is via its intersection theory, the study of how certain loci in the moduli space (those corresponding to curves with specific properties) intersect each other. In this project, the PI will investigate various problems relating to a fundamental intersection-theoretic class on the moduli space of curves: the double ramification cycle. This project will also provide research opportunities for graduate students, who will be trained in the methods of the field.The PI will study two main groups of problems dealing with the intersection theory of the moduli space of curves. First, the PI will develop improved formulas for the logarithmic double ramification cycle, a refinement of the double ramification cycle constructed and studied in the last several years by Holmes, Ranganathan, Schwarz, and others. The PI and his coauthors recently developed an approach to computing this cycle, and the primary goal of this project is to make this approach more effective so that it can be applied to localization computations in logarithmic geometry. Second, the PI will investigate assorted connections that have surfaced in recent years between the double ramification cycle and other tautological classes. This includes developing a theory of log tautological relations, defining and computing an orbifold version of the double ramification cycle, and attempting to prove a one-dimensional socle result for the tautological ring of the moduli space of bridgeless curves.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.

代数几何是研究由多项式方程定义的空间的几何性质的数学分支。曲线模空间是代数几何中的一个重要对象，它是一个几何空间，其点对应于代数曲线的类型。这个模空间与数学和物理的其他领域有许多联系（其中代数曲线是弦理论中出现的弦）。研究曲线模空间的一种方法是通过其相交理论，研究模空间中的某些轨迹（与具有特定性质的曲线相对应的轨迹）如何相互相交。在这个项目中，PI将研究与曲线模空间上的基本相交理论类有关的各种问题：双分枝循环。该项目还将为研究生提供研究机会，他们将接受该领域方法的培训。PI将研究处理曲线模空间的相交理论的两组主要问题。首先，PI将开发对数双分枝循环的改进公式，这是Holmes、Ranganathan、Schwarz等人在过去几年中构建和研究的双分枝循环的改进。PI和他的合作者最近开发了一种计算这个周期的方法，这个项目的主要目标是使这种方法更有效，以便它可以应用于对数几何中的定位计算。其次，PI将调查近年来出现的双分支循环和其他同义类之间的各种联系。这包括发展对数同义关系的理论，定义和计算双分支循环的一个轨道版本，并试图证明无桥曲线模空间的同义环的一维社会结果。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。