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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，研究对称，特别是线性数学结构的对称，使用可逆矩阵群。另一方面，代数几何是基于使用抽象的代数技术，主要来自交换代数，来解决关于多变量多项式的零集的几何问题。本NSF奖支持的研究处于表征理论与代数几何的交叉点，用表征理论解决几何问题，用几何方法理解表征理论。研究项目主要围绕两个主题展开。首先，PI将发展曲线上一般抛物面群方案的共形块理论。重点研究扭曲共形块的Pappas-Rapoport猜想、消失猜想和Verlinde公式。这将导致在轨道共形场理论和几何朗兰兹规划中的应用。对于第二个主题，PI将发展扭曲仿射格拉斯曼几何与表征理论之间的联系。仿射Grassmannians的几何可以通过几何Satake对应与Kac-Moody理论、约化群的表示理论及其基联系起来，在辛对偶中也起着至关重要的作用。沿着这一方向的研究将推进扭曲仿射Grassmannians在Kac-Moody理论、分支几何Satake、辛对偶以及与对称空间施普林格理论的联系中的作用。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。