Moduli Spaces, Tautological Rings, and Theta Functions

该项目解决了代数几何中的问题。代数几何是研究多变量多项式方程解集的学科。它是数学中最古老、最发达和最活跃的研究领域之一。它与数学中的许多子领域，无论是纯粹的还是应用的，以及计算机科学和理论物理都有联系。模理论关注几何形状的行为，因为定义方程的系数是允许变化的。这个项目的重点是复杂曲面的模量。这些空间在过去被抽象地研究过。该项目将通过理解几何计算中出现的曲面模空间内的自然轨迹集及其结构来进一步研究这些空间。特别是，结合经典代数几何和理论物理的工具，该项目旨在开发一种灵活的机制来定量地处理这些位点，并进一步应用该机制来回答曲面枚举几何中出现的问题（即：枚举所有曲面具有某些几何性质的问题）。除了研究之外，首席研究员还将参与高中数学竞赛，为本科生、研究生和博士后提供建议，努力增加数学的多样性，并组织研讨会和地方会议。具体而言，该项目涉及曲面（K3曲面，阿贝尔曲面等）模的重言环的研究。这里的一些项目涉及获得结构结果，了解同义环的大小，以及发展同义类的显式演算，自然几何循环在同义类中的表达式。这也导致了独立的兴趣项目，例如，关于Hilbert格式上的重言积分的Lehn猜想的推广，高阶稳定对对应，以及与新曲面类的Verlinde公式的联系。在曲线、K3或阿贝尔曲面的模上，研究者将继续研究广义函数束。这里的问题涉及精练的陈氏文字计算和与同义环的联系。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。

The project addresses questions in algebraic geometry. Algebraic geometry concerns the study of solution sets of polynomial equations several variables. It is one of the oldest, most developed and active areas of research in mathematics. It has connections to many sub-fields within mathematics, both pure and applied, and also to computer science and theoretical physics. Moduli theory concerns the behavior of geometric shapes as the coefficients of the defining equations are allowed to vary. The focus of this project is the moduli of complex surfaces. These spaces have been studied abstractly in the past. This project will further study these spaces by understanding the set of natural loci inside the moduli space of surfaces that arise in geometric computations, as well as its structure. In particular, combining tools from classical algebraic geometry as well as theoretical physics, the project aims to develop a flexible machinery to work quantitatively with these loci, and to further apply this machinery to answer questions arising in the enumerative geometry of surfaces (that is, questions of the type: enumerate all surfaces that enjoy certain geometric properties). In addition to research, the principal investigator will be involved in high school mathematics competitions, will advise undergraduate students, graduate students, and postdoctoral associates, work to increase diversity in mathematics, and organize seminars and local conferences.Specifically, the project concerns the study of the tautological ring of the moduli of surfaces (K3 surfaces, abelian surfaces, etc.). Some of the projects here concern obtaining structural results, understanding the size of the tautological rings, and developing an explicit calculus for tautological classes, expressions for natural geometric cycles in terms of tautological classes. This also leads to projects of independent interest, e.g., generalizations of a conjecture of Lehn concerning tautological integrals over the Hilbert scheme, a higher-rank stable pair correspondence, and connections with the Verlinde formula for new classes of surfaces. Over the moduli of curves, K3 or abelian surfaces, the investigator will continue work on bundles of generalized theta functions. Questions here deal with refined Chern character calculations and connections with the tautological rings.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.