Double Ramification Cycles and Tautological Classes

代数几何是数学的一个分支，研究由多项式方程定义的空间的几何性质。代数几何中的一个关键对象是曲线的模空间，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学的其他领域有许多联系（代数曲线是弦理论中出现的弦）。研究曲线的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（对应于具有特定属性的曲线）如何相互相交。在这个项目中，PI将研究与曲线的模空间上的基本相交理论类相关的各种问题：双重分歧循环。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将研究两组主要问题，涉及曲线模空间的相交理论。首先，PI将开发对数双分歧循环的改进公式，这是霍姆斯、阮加纳森、施瓦茨和其他人在过去几年中构建和研究的双分歧循环的改进。PI和他的合著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局部化计算。第二，PI将研究近年来在双分歧循环和其他重言式类之间出现的各种联系。这包括发展对数重言式关系的理论，定义和计算双重分歧循环的轨道版本，并试图证明无桥曲线的模空间的重言式环的一维柱脚结果。该奖项反映了NSF的法定使命，并被认为值得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来支持。

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.