Conformal Blocks and Affine Grassmannian Associated to Parahoric Group Schemes

共形块和仿射格拉斯曼与超视群方案相关

基本信息

批准号：
2001365
负责人：
Jiuzu Hong
金额：
$ 15.5万
依托单位：
University of North Carolina at Chapel Hill
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-06-01 至 2024-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001365&HistoricalAwards=false
关键词：
Conformal Blocks Affine Grassmannian Associated

项目摘要

Representation theory is a branch of algebra studying symmetries, especially symmetries of linear mathematical structures, using groups of invertible matrices. On the other hand, algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometric problems about the sets of zeros of multivariable polynomials. The research supported by this NSF award is at the crossroads of representation theory and algebraic geometry, to use representation theory to solve geometric problems and to use geometric methods to understand representation theory.The research projects center on two main themes. In the first, the PI will develop a theory of conformal blocks for general parahoric group schemes over curves. The research will focus on Pappas-Rapoport conjecture, vanishing conjecture and Verlinde formula for twisted conformal blocks. This will lead to applications in orbifold conformal field theory and geometric Langlands program for parahoric group schemes. For the second theme, the PI will develop connections between the geometry of twisted affine Grassmannian and representation theory. The geometry of affine Grassmannians can be related to Kac-Moody theory, representation theory of reductive groups and their bases via geometric Satake correspondence, and it also plays crucial role in symplectic duality. The research along this direction will advance the role of twisted affine Grassmannians in Kac-Moody theory, ramified geometric Satake, symplectic duality, and a connection with Springer theory for symmetric spaces.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.

表示论是代数的一个分支，利用可逆矩阵群来研究对称性，特别是线性数学结构的对称性。另一方面，代数几何基于抽象代数技术（主要来自交换代数）的使用，用于解决有关多元多项式零点集的几何问题。该NSF奖项支持的研究处于表示论和代数几何的十字路口，用表示论解决几何问题，用几何方法理解表示论。研究项目集中在两个主题上。首先，PI 将为曲线上的一般旁斜群方案开发共角块理论。研究重点是Pappas-Rapoport猜想、消失猜想和扭曲共形块的Verlinde公式。这将导致在轨道共形场理论和旁群方案的几何朗兰兹程序中的应用。对于第二个主题，PI 将发展扭曲仿射格拉斯曼几何与表示理论之间的联系。仿射格拉斯曼几何可以通过几何Satake对应与Kac-Moody理论、还原群表示论及其基联系起来，并且在辛对偶性中也起着至关重要的作用。沿着这个方向的研究将推进扭曲仿射格拉斯曼函数在 Kac-Moody 理论、分支几何 Satake、辛对偶性以及与对称空间的 Springer 理论的联系中的作用。该奖项反映了 NSF 的法定使命，并被认为值得通过以下方式获得支持：使用基金会的智力价值和更广泛的影响审查标准进行评估。