Moduli Spaces, Tautological Rings, and Theta Functions

该项目涉及代数几何问题。代数几何研究多元多项式方程的解集.它是数学研究中最古老、最发达和最活跃的领域之一。它与数学中的许多子领域都有联系，包括纯数学和应用数学，以及计算机科学和理论物理。模量理论关注的是几何形状的行为，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表面的模量。这些空间在过去已经被抽象地研究过了。这个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，结合工具，从经典代数几何以及理论物理，该项目旨在开发一个灵活的机器定量工作与这些轨迹，并进一步应用这种机器来回答问题中出现的枚举几何的表面（即，问题的类型：枚举所有表面享有某些几何属性）。除了研究之外，主要研究者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，组织研讨会和地方会议。具体来说，该项目涉及曲面（K3曲面，阿贝尔曲面等）模的重言式环的研究。这里的一些项目涉及获得结构的结果，了解重言环的大小，并制定一个明确的演算重言类，表达自然几何周期的重言类。这也导致了独立感兴趣的项目，例如，概括的猜想莱恩关于重言式积分的希尔伯特计划，一个高秩稳定对对应，并连接与Verlinde公式的新类别的表面。在曲线，K3或阿贝尔曲面的模量，研究人员将继续工作的广义θ函数束。这个奖项反映了NSF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.