Analytic Langlands Correspondence

这是一个与数论和弦论有联系的代数几何领域的项目。代数几何是研究由多项式方程和相关数学结构定义的几何对象的学科。将进行三个研究项目。在主要项目中，PI将提供自同构形式理论的推广，这是数论中一个重要的经典有根领域。本项目为研究生提供研究训练机会。更详细地说，主要项目将有助于局部场上曲线的解析朗兰兹对应。目的是研究Hecke算子在Schwartz密度空间上与局部场上曲线束的模堆相关的作用，并将相关的特征函数和特征值与具有相应伽罗瓦群作用的对象联系起来。作为该项目的一部分，PI将证明在稳定束和非常稳定束对应的点附近的束堆上的施瓦茨密度行为的结果。第二个项目与稳定超曲线的几何有关。该PI将为超曲线模空间的超弦超测度的积分奠定坚实的基础。第三个项目是由穿孔球的对称幂的同调镜像对称驱动的：PI将在与某些环面超表面奇点的环面分辨率相关的范畴上构造各种映射类群的作用，并将找到这个图与Ozsvath-Szabo的绝对结不变量的关系。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。

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.