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

20世纪数学中最重要的进步之一是观点的改变：我们应该扩大我们的视野，研究不同类别的对象是如何在家庭中组合在一起的，而不是研究单一的数学对象。用生态学来类比，这种视角的变化类似于意识到，为了了解一条鱼在海洋中的运动，了解这条鱼如何与其鱼群中的其他成员互动非常有帮助。在数学中，模数空间的概念大致指的是整个物体家族；例如，在鱼的类比中，模数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，我们可以通过研究将它们参数化的模空间的形状来了解我们感兴趣的对象。了解其边界附近的模空间的形状尤其具有启发性，而这项由NSF奖支持的研究的目标是了解一些模空间的边界的形状，这些模空间将不同类型的代数曲线族参数化。这个项目为本科生和研究生提供了研究培训的机会。这个项目的研究内容分为三个相互关联的类别，都有一个共同的主题，即研究各种紧致的曲线模空间，以及从它们的边界结构中可以收集到哪些几何和计数信息。在第一行问题中，PI将研究新的模空间类，这些类可以实现为与某些复杂反射群相关的奇妙紧化。这些新的模空间提供了一个肥沃的试验场，用于研究多面体方法在多大程度上可以推广到环面变量之外。在第二行问题中，PI将把模空间引入复反射群中的因式分解问题的研究。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因式分解的多项式结构。在最后一系列问题中，PI将开始研究伪稳定曲线的模空间的重言环。这些空间提供了曲线的模空间的替代紧致，允许具有尖端奇点的曲线，而不是通常的节点奇点，这项研究的进展将导致关于具有尖端奇点的曲线的计数几何的进展。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。

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.