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Moduli Spaces, Tautological Rings, and Theta Functions

模空间、同义反复环和 Theta 函数

基本信息

批准号：
1802228
负责人：
Dragos Oprea
金额：
$ 26.5万
依托单位：
University of California-San Diego
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1802228&HistoricalAwards=false
关键词：
Moduli Spaces Tautological Rings Theta

项目摘要

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.

该项目解决了代数几何中的问题。代数几何是研究多元多项式方程的解集的学科。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有联系，既有纯粹的，也有应用的，也有计算机科学和理论物理的。当定义方程的系数被允许变化时，模理论涉及几何形状的行为。本项目的重点是复杂曲面的模数计算。过去，人们对这些空间进行了抽象的研究。这个项目将通过了解几何计算中出现的曲面的模空间中的自然轨迹集以及它的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器来定量地处理这些轨迹，并进一步应用该机器来回答曲面的计数几何中出现的问题(即，类型的问题：列举所有具有某些几何性质的曲面)。除了研究，首席研究员还将参与高中数学竞赛，为本科生、研究生和博士后助理提供建议，努力增加数学的多样性，并组织研讨会和本地会议。具体地说，该项目涉及曲面(K3曲面、阿贝尔曲面等)的模重言环的研究。这里的一些项目涉及获得结构结果，了解重言式环的大小，以及开发重言式类的显式演算，以重言式类的形式表示自然几何循环。这也导致了独立感兴趣的项目，例如，关于Hilbert格式上的重言式积分的Lehn猜想的推广，更高阶稳定对对应，以及与新的曲面类的Verlinde公式的联系。在曲线、K3或阿贝尔曲面的模数上，研究人员将继续研究广义theta函数丛。这里的问题涉及到陈冠希性格的精确计算和与同义反复环的联系。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

期刊论文数量（6）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Euler characteristics of tautological bundles over Quot schemes of curves

曲线 Quot 格式上同义反复丛的欧拉特征

DOI：
10.1016/j.aim.2023.108943
发表时间：
2023
期刊：
Advances in Mathematics
影响因子：
1.7
作者：
Oprea, Dragos;Sinha, Shubham
通讯作者：
Sinha, Shubham

The virtual K-theory of Quot schemes of surfaces

曲面Quot方案的虚拟K理论

DOI：
10.1016/j.geomphys.2021.104154
发表时间：
2021
期刊：
Journal of Geometry and Physics
影响因子：
1.5
作者：
Arbesfeld, Noah;Johnson, Drew;Lim, Woonam;Oprea, Dragos;Pandharipande, Rahul
通讯作者：
Pandharipande, Rahul

Higher rank Segre integrals over the Hilbert scheme of points