Asymptotic Hodge Theory, Fibered Motives, and Algebraic Cycles

渐近霍奇理论、纤维动机和代数圈

• 批准号: 2101482
• 负责人: Matthew Kerr
• 金额: $ 16.48万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-08-15 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101482&HistoricalAwards=false
• 关键词: Asymptotic; Hodge; Theory; Fibered; Motives

The focus of this project is on polynomial equations and their solution sets, which have been studied specifically and intensively by algebraic geometers. Deep conjectures, including those of Bloch, Beilinson and Hodge, seek to relate the behavior of analytic objects (like integrals and differential equations), which are a priori non-algebraic, to the underlying algebraic structure and topological shape of such solution sets. Even as these conjectures remain intractable in general, solutions in individual cases both continue to bear out their validity and produce new algebraic structures (for example "cycles" and "motives") which facilitate the solutions of problems in apparently remote areas of mathematics and other sciences, for example, at the interface of number theory and physics (such as string theory and quantum field theory). Recent technical innovations, based on considering polynomial equations in families, have begun to provide access to new cases of these conjectures. Their further development and application is the subject of this project, whose results will be disseminated through conferences, summer schools, journal articles and websites. The project consultants brought to Washington University by the grant will contribute to its research atmosphere, and specialized problems embedded in the project will provide training for the PI's graduate students.Hodge-theoretic invariants such as period and regulator maps provide the basic interface between the algebraic and transcendental worlds in modern geometry. The goal of this project is to better understand the asymptotic properties of these invariants, and apply the results to closely intertwined problems of current interest in arithmetic geometry, physics, and algebraic geometry. Specifically, the PI plans to: (I) exhibit the Apery constants of Fano varieties as limits of higher normal functions (hence periods), and construct motives related to motivic Gamma functions to verify specific instances of conjectures of Beilinson and Green-Griffiths-Kerr; (II) compute the Feynman amplitudes associated to a family of two-loop graphs, and relate the spectra of quantum curves to zeroes and limits of normal functions (thereby confirming two consequences of a conjecture of Marino); and (III) use the mixed Hodge theory of miniversal deformations of singularities to interpret fibrations of geometric boundary components in moduli, and use Lie-theoretic methods to study local and global aspects of Hodge-theoretic compactifications of period maps.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.

该项目的重点是多项式方程及其解决方案集，这些方程已通过代数几何图是特定和深入研究的。深层猜想，包括Bloch，Beilinson和Hodge的猜想，试图将分析对象的行为（例如积分和微分方程）（如先验的非代数）与此类溶液集的基本代数结构和拓扑形状联系起来。即使这些猜想一般而言，在个别情况下的解决方案仍然保持有效性，并产生新的代数结构（例如“循环”和“动机”），从而促进了在数学和其他科学方面偏远地区的问题解决方案，例如，在数字理论和物理学的界面理论和量子理论和量子理论和量子理论和量子理论理论和量子理论理论和量子理论）上。基于考虑家庭中多项式方程的最新技术创新已开始提供这些猜想的新案例。他们的进一步开发和应用是该项目的主题，该项目的结果将通过会议，暑期学校，期刊文章和网站传播。该项目顾问将通过赠款带给华盛顿大学的研究氛围，嵌入该项目的专业问题将为PI的研究生提供培训。Hodge理论不变式（例如时期和监管图）提供了代数和现代报道中代数和超越世界之间的基本界面。该项目的目的是更好地了解这些不变性的渐近特性，并将结果应用于当前在算术几何，物理和代数几何学上的兴趣紧密相互交织的问题。具体而言，PI计划：（i）以较高的正常功能（因此时期）的限制表现出FANO品种的Apery常数，并构建与动机伽马功能相关的动机，以验证贝林森和绿色 - 绿色 - 帝国 - 克里夫斯 - 凯尔的特定实例； （ii）计算与两环图家族相关的Feynman振幅，并将量子曲线的光谱与正常函数的零和限制相关联（从而确认了Marino的猜想的两个后果）； （iii）使用奇异性的微型变形的混合杂货理论来解释模数中几何边界组成部分的纤维化，并使用撒谎理论方法来研究周期映射的Hodge理论压缩的本地和全球方面。这些奖项的奖项奖励NSF的法定任务，反映了综述的综述，这是通过评估的范围来进行的。

项目成果

Algebraic and Analytic Compactifications of Moduli Spaces

模空间的代数和解析紧致化

• DOI: 10.1090/noti2541
• 发表时间: 2022
• 期刊: Notices of the American Mathematical Society
• 作者: Gallardo, Patricio;Kerr, Matt
• 通讯作者: Kerr, Matt

Deformation of rational singularities and Hodge structure

• DOI: 10.14231/ag-2022-014
• 发表时间: 2019-06
• 期刊: Algebraic Geometry
• 影响因子: 1.5
• 作者: M. Kerr;R. Laza;M. Saito

Unipotent extensions and differential equations (after Bloch–Vlasenko)

单能扩张和微分方程（仿布洛赫·弗拉森科）

• DOI: 10.4310/cntp.2022.v16.n4.a5
• 发表时间: 2022
• 期刊: Communications in Number Theory and Physics
• 影响因子: 1.9
• 作者: Kerr, Matt