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RUI: Compactifying Moduli Spaces of Orbits, Covers, and Curves

RUI：压缩轨道、覆盖和曲线的模空间

基本信息

批准号：
2001439
负责人：
Dustin Ross
金额：
$ 16.02万
依托单位：
San Francisco State University
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：
2020-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001439&HistoricalAwards=false
关键词：
RUI Compactifying Moduli Spaces Orbits

项目摘要

One of the most consequential advances in mathematics during the 20th century was motivated by a change in perspective: instead of studying a single mathematical object, we should broaden our scope and study how classes of objects fit together in families. To draw an analogy with ecology, this change in perspective is akin to the realization that, in order to understand the movement of a single fish in the sea, it helps a great deal to understand how that fish interacts with the other members of their school. In mathematics, the notion of a moduli space loosely refers to an entire family of objects; for example, in the fish analogy, the moduli space could refer to the entire school of fish. Moduli spaces can, themselves, be treated as a single entity, comprised of many, and we can learn about the objects we are interested in by studying the shape of the moduli space that parametrizes them. It can be especially enlightening to understand the shape of moduli spaces near their boundary, and the research supported by this NSF award is driven by the goal of understanding the shape of the boundary of a number of moduli spaces that parametrize different types of families of algebraic curves. This project provides research training opportunities for undergraduate and graduate students.The research aspects in this project fall into three interrelated categories, all with the common theme of investigating various compact moduli spaces of curves and what geometric and enumerative information can be gleaned from the structure of their boundary. In the first line of problems, the PI will study new classes of moduli spaces that can be realized as wonderful compactifications associated to certain complex reflection groups. These new moduli spaces provide a fertile testing ground for investigating the extent to which polyhedral methods can be generalized beyond toric varieties. In the second line of problems, the PI will introduce moduli spaces into the study of factorization problems in complex reflection groups. In particular, the primary objective is to study the polynomial structure of factorizations by constructing a suitable compactification of the associated moduli spaces of admissible covers. In the final line of problems, the PI will initiate a study of the tautological rings of the moduli spaces of pseudo-stable curves. These spaces provide alternative compactifications of the moduli spaces of curves that allow for curves with cuspidal singularities, instead of the usual nodal singularities, and progress in this research would lead to advances concerning the enumerative geometry of curves with cuspidal singularities.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.

20世纪数学最重要的进步之一是由观点的改变所推动的：我们不应该研究单一的数学对象，而应该扩大我们的范围，研究各种类型的对象如何在家庭中相互配合。用生态学来类比，这种观点的改变类似于认识到，为了理解海洋中一条鱼的运动，理解这条鱼是如何与鱼群中的其他成员互动的，这对理解这条鱼有很大帮助。在数学中，模空间的概念泛指一整个对象族；例如，在鱼的类比中，模空间可以指整个鱼群。模空间本身，可以被视为一个单一的实体，由许多个体组成，我们可以通过研究参数化它们的模空间的形状来了解我们感兴趣的对象。理解模空间边界附近的形状尤其具有启意义，NSF资助的这项研究的目标是理解一些参数化不同类型代数曲线族的模空间的边界形状。本项目为本科生和研究生提供研究训练机会。这个项目的研究方面分为三个相互关联的类别，所有这些都有一个共同的主题，即研究曲线的各种紧模空间，以及从它们的边界结构中可以收集到什么几何和枚举信息。在第一行问题中，PI将研究新的模空间类，这些模空间可以被实现为与某些复杂反射群相关的奇妙紧化。这些新的模空间为研究多面体方法可以推广到何种程度提供了肥沃的试验田。在第二行问题中，PI将把模空间引入到复反射群分解问题的研究中。特别地，主要目的是通过构造可容许覆盖的相关模空间的适当紧化来研究分解的多项式结构。在最后的问题中，PI将开始研究伪稳定曲线模空间的重言环。这些空间提供了曲线模空间的可选紧化，允许曲线具有尖角奇点，而不是通常的节点奇点，该研究的进展将导致具有尖角奇点曲线的计数几何的进展。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。