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

20世纪数学最重要的进步之一是由视角的改变所推动的：我们不应该研究单一的数学对象，而应该扩大我们的范围，研究各类对象如何在家庭中组合在一起。用生态学来类比，这种观点的变化类似于认识到，为了了解海洋中一条鱼的运动，了解这条鱼如何与鱼群中的其他成员互动会有很大帮助。在数学中，模空间的概念大致上指的是整个对象家族;例如，在鱼的类比中，模空间可以指的是整个鱼群。模空间本身可以被视为一个单一的实体，由许多组成，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。理解模空间在其边界附近的形状特别有启发性，并且由该NSF奖项支持的研究是由理解参数化不同类型的代数曲线族的一些模空间的边界形状的目标驱动的。本计画为本科生及研究生提供研究训练机会，研究内容分为三个相关的范畴，其共同主题为探讨曲线的各种紧致模空间，以及从其边界结构中可获得几何及计数资讯。在第一行的问题，PI将研究新的类模空间，可以实现为美妙的紧化与某些复杂的反射群。这些新的模空间提供了一个肥沃的试验田调查多面体的方法可以推广到环面品种以外的程度。在问题的第二行，PI将引入模空间到复反射群的因子分解问题的研究中。特别是，主要目标是通过构造一个合适的紧化的相关模空间的容许覆盖，研究多项式结构的分解。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式环。这些空间提供了曲线的模空间的另一种紧化，允许曲线具有尖点奇点，而不是通常的节点奇点，这项研究的进展将导致尖点奇点曲线的枚举几何学的进步。该奖项反映了美国国家科学基金会的法定使命，并被认为值得通过利用基金会的智力价值和更广泛的影响审查进行评估来支持的搜索.

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.