Analytic Langlands Correspondence

分析朗兰兹通讯

• 批准号: 2349388
• 负责人: Alexander Polishchuk
• 金额: $ 25.52万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2349388&HistoricalAwards=false
• 关键词: Analytic; Langlands; Correspondence

This is a project in the field of algebraic geometry with connections to number theory and string theory. Algebraic geometry is the study of geometric objects defined by polynomial equations, and related mathematical structures. Three research projects will be undertaken. In the main project the PI will provide a generalization of the theory of automorphic forms, which is an important classical area with roots in number theory. This project provides research training opportunities for graduate students. In more detail, the main project will contribute to the analytic Langlands correspondence for curves over local fields. The goal is to study the action of Hecke operators on a space of Schwartz densities associated with the moduli stack of bundles on curves over local fields, and to relate the associated eigenfunctions and eigenvalues to objects equipped with an action of the corresponding Galois group. As part of this project, the PI will prove results on the behavior of Schwartz densities on the stack of bundles near points corresponding to stable and very stable bundles. A second project is related to the geometry of stable supercurves. The PI will develop a rigorous foundation for integrating the superstring supermeasure of the moduli space of supercurves. The third project is motivated by the homological mirror symmetry for symmetric powers of punctured spheres: the PI will construct the actions of various mapping class groups on categories associated with toric resolutions of certain toric hypersurface singularities and will find a relation of this picture to Ozsvath-Szabo's categorical knot invariants.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.

这是一个代数几何领域的项目，与数论和弦理论有关。代数几何是研究由多项式方程定义的几何对象和相关的数学结构。将开展三个研究项目。在主要项目中，PI将提供自守形式理论的推广，自守形式是数论中一个重要的经典领域。该项目为研究生提供了研究培训机会。 更详细地说，主要项目将有助于局部域上曲线的解析朗兰兹对应。目标是研究Hecke算子在Schwartz密度空间上的作用，该空间与局部域上曲线上的丛的模堆叠相关联，并将相关的本征函数和本征值与配备有相应伽罗瓦群作用的对象相关联。作为该项目的一部分，PI将证明在对应于稳定和非常稳定束的点附近的束堆上的Schwartz密度的行为。第二个项目是关于稳定超曲线的几何。PI将为整合超曲线模空间的超弦超测度奠定严格的基础。第三个项目的动机是同调镜像对称的对称功率的穿孔领域：PI将在与某些复曲面超曲面奇点的复曲面分辨率相关联的类别上构造各种映射类组的作用，并将找到该图片与Ozsvath的关系。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的学术价值和更广泛的影响审查标准。