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 世纪数学领域最重要的进步之一是由观点的变化推动的：我们不应该研究单个数学对象，而应该扩大我们的范围并研究对象类别如何在家庭中组合在一起。与生态学进行类比，这种观点的变化类似于这样的认识：为了理解海中一条鱼的运动，它有助于理解该鱼如何与鱼群中的其他成员相互作用。。在数学中，模空间的概念松散地指代整个对象族。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由多个实体组成的单个实体，我们可以通过研究参数化对象的模空间的形状来了解我们感兴趣的对象。了解模空间边界附近的形状尤其具有启发性，这项 NSF 奖项支持的研究是由了解多个模空间边界形状的目标驱动的，这些模空间对不同类型的代数族进行参数化曲线。该项目为本科生和研究生提供研究培训机会。该项目的研究方面分为三个相互关联的类别，所有类别的共同主题是研究曲线的各种紧模空间以及可以从结构中收集哪些几何和枚举信息他们的边界。在第一行问题中，PI 将研究新类别的模空间，这些模空间可以实现为与某些复杂反射群相关的精彩紧化。这些新的模空间为研究多面体方法可以推广到复曲面簇之外的程度提供了肥沃的试验场。在第二行问题中，PI将把模空间引入到复杂反射群因式分解问题的研究中。特别是，主要目标是通过构建可接受覆盖的相关模空间的适当压缩来研究因式分解的多项式结构。在最后一行问题中，PI 将启动对伪稳定曲线模空间的同义反复环的研究。这些空间提供了曲线模空间的替代紧化，允许曲线具有尖点奇点，而不是通常的节点奇点，并且这项研究的进展将导致有关具有尖点奇点的曲线的枚举几何的进步。该奖项反映了 NSF 的法定使命，并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，被认为值得支持。