Geometric Methods in Representation Theory and the Langlands Program

表示论中的几何方法和朗兰兹纲领

• 批准号: 2101837
• 负责人: Charlotte Chan
• 金额: $ 24万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101837&HistoricalAwards=false
• 关键词: Geometric; Methods; Representation; Theory; Langlands

Representation theory is the study of symmetries using linear algebra. As such, it is deeply intertwined with a vast range of mathematical fields and has rich applications, to particle physics, physical chemistry, and computer vision for example. Since the dawn of the subject nearly a century ago, interactions between representation theory and algebraic geometry, which is the study of solutions of polynomial equations, have influenced the landscape of each area. More recently, in the past several decades, number theory has played an increasing important role in this world. The Langlands program is a far-reaching web of conjectures which predicts unexpected matchings between deep number-theoretic and representation-theoretic phenomena. This project includes opportunities for student research. This project is centered on geometric methods in representation theory within the framework of the Langlands program. The long-term objective is to construct a geometric theory of representations of p-adic groups. This will allow passage between rapid developments in the traditionally algebraic approach to representations of p-adic groups and rapid developments in geometric progress in the Langlands correspondence. Moreover, it will relate conjectural algebraic constructions of L-packets to deep geometric phenomena.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.

表示论是使用线性代数研究对称性的学科。因此，它与广泛的数学领域紧密相连，并具有丰富的应用，例如粒子物理、物理化学和计算机视觉。自近一个世纪前该学科诞生以来，表示论和代数几何（研究多项式方程的解）之间的相互作用已经影响了每个领域的景观。最近，在过去的几十年里，数论在这个世界上发挥了越来越重要的作用。朗兰兹纲领是一个影响深远的猜想网，它预测了深层数论和表示论现象之间的意外匹配。该项目包括学生研究的机会。该项目以朗兰兹计划框架内的表示论几何方法为中心。长期目标是构建 p 进群表示的几何理论。这将允许在 p-adic 群表示的传统代数方法的快速发展和朗兰兹对应中几何级数的快速发展之间进行过渡。此外，它将 L 包的猜想代数构造与深层几何现象联系起来。该奖项反映了 NSF 的法定使命，并通过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，被认为值得支持。