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.

表示论是研究代数中的对称性。对对称性的理解使我们能够把复杂的问题简化为简单的问题。 代数可以用来描述贯穿数学和真实的世界的各种现象和结构，因此表示论有许多重要的应用。层是推广函数的通常概念的几何对象，并且已被证明在推进我们对表示论的理解方面非常有效。这个研究项目旨在揭示层的更精细的信息以及这些信息在表示论中的应用。宇称层是由PI及其合作者引入的，作为研究具有正特征的约化群的表示论的工具。 宇称层的研究也表明了反常层范畴中新结构的存在。 PI将在一些特殊的，重要的情况下探索这些结构，以及它们在许多领域的预期应用，包括Hecke代数的表示和Lie型有限群的模表示。要研究的几何空间是幂零锥及其对对称对和规范理论的推广，以及（广义）旗簇和环面簇。建议的方法包括利用上同调宇称消失属性，附近的循环和双曲本地化functors.This奖反映了NSF的法定使命，并已被认为是值得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的支持。