Categories of Sheaves in Representation Theory

表示论中滑轮的类别

• 批准号: 1802299
• 负责人: Carl Mautner
• 金额: $ 16.71万
• 依托单位: University of California-Riverside
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1802299&HistoricalAwards=false
• 关键词: Categories; Sheaves; Representation; Theory

Representation theory is the study of symmetries in algebra. An understanding of symmetry allows us to reduce complicated problems to simpler ones. Algebra can be used to describe a wide range of phenomena and structures throughout mathematics and the real world, and consequently representation theory has many important applications. Sheaves are geometric objects that generalize the usual notion of functions and have proven to be extremely effective in advancing our understanding of representation theory. This research project aims to uncover finer information about sheaves and applications of this information to representation theory.Parity sheaves were introduced by the PI and his collaborators as a tool for studying the representation theory of reductive groups in positive characteristic. The study of parity sheaves has also suggested the existence of new structures in categories of perverse sheaves. The PI will explore these structures in some special, important, cases and their expected applications in a number of areas including the representations of Hecke algebras and modular representations of finite groups of Lie type. The geometric spaces to be studied are nilpotent cones and their generalizations for symmetric pairs and in gauge theory, as well as (generalized) flag varieties and toric varieties. The proposed methods include utilizing cohomological parity vanishing properties, nearby cycles and hyperbolic localization functors.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.

表示理论是对代数中对称性的研究。对对称性的理解使我们能够将复杂的问题减少到简单的问题上。 代数可用于描述数学和现实世界中的各种现象和结构，因此表示论具有许多重要的应用。滑轮是几何对象，它概括了通常的功能概念，并且被证明在促进我们对表示理论的理解方面非常有效。该研究项目旨在揭示有关滑轮的更精细信息以及这些信息在表示论中的应用。宇称滑轮是由 PI 及其合作者引入的，作为研究正特性还原群表示论的工具。 对奇偶校服的研究还表明，在不正正滑轮类别中存在新结构。 PI将在一些特殊的，重要的案例及其在许多领域的预期应用中探索这些结构，包括Hecke代数的表示以及有限的LIE类型组的模块化表示。要研究的几何空间是nilpotent锥及其对称对，量规理论的概括，以及（概括性的）标志品种和曲折品种。所提出的方法包括利用共同的奇偶校验消失的特性，附近的周期和双曲线定位函数。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估评估的评估来支持的。