Geometric and Cohomological Invariants in Modular Representation Theory

模表示理论中的几何和上同调不变量

• 批准号: 1501146
• 负责人: Julia Pevtsova
• 金额: $ 23.99万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1501146&HistoricalAwards=false
• 关键词: Geometric; Cohomological; Invariants; Modular; Representation

Representation theory studies actions of groups and other algebraic structures on vector spaces. It has its origins in the study of symmetries and emerged as a subject in its own right more than a hundred years ago in the work of Frobenius and Schur. In its current stage of development, representation theory is intimately intertwined with numerous branches of mathematics, such as geometry, topology, combinatorics, and analysis, as well as with physics. In this project the PI will establish new connections between representation theory and geometry, and develop techniques that will shed light on longstanding problems in both areas. The project also contains several educational outreach initiatives aimed at elementary, middle and junior high school student in Seattle area. These initiatives range from an enrichment math program at a local elementary school, to a city wide math circle for middle schoolers, to public math lectures and a special Math Olympiad for students in grades 5-10. These opportunities are designed to attract more young people, particularly young women, to careers in the mathematical sciences, and to raise awareness and appreciation of the beautiful nature of mathematics in the next generation.This research focuses on four interrelated directions traceable to Quillen's fundamental work in group cohomology from 1971. A significant part of the project concentrates on interpreting categories associated to representations in geometric terms. In the first three projects, the PI will classify triangulated subcategories using support theories, find an analogue of the Beilinson-Bernstein localization theorem for complex Lie algebras in the case of infinitesimal neighborhoods of algebraic groups in positive characteristic, and investigate the Orlov correspondence for representation theoretic categories. In a fourth project, which is of a more algebraic nature and has geometric applications, the PI will establish the finite generation of cohomology for a new class of finite dimensional Hopf algebras.

表示理论研究群和其他代数结构在向量空间上的作用。它起源于对对称的研究，并在一百多年前Frobenius和Schur的工作中成为一门独立的学科。在目前的发展阶段，表示理论与数学的许多分支，如几何学、拓扑学、组合学和分析学以及物理学紧密地交织在一起。在这个项目中，PI将在表示理论和几何之间建立新的联系，并开发技术，以阐明这两个领域长期存在的问题。该项目还包括针对西雅图地区小学、初中和初中学生的几项教育推广活动。这些举措包括在当地一所小学开设数学丰富课程，在全市范围内为中学生开设数学圈，以及为5-10年级的学生举办公开数学讲座和特别的数学奥林匹克竞赛。这些机会旨在吸引更多的年轻人，特别是年轻女性，从事数学科学事业，并提高下一代对数学美丽本质的认识和欣赏。本研究的重点是四个相互关联的方向，可追溯到Quillen从1971年开始在群上同论方面的基础工作。该项目的一个重要部分集中在解释与几何术语表示相关的类别。在前三个项目中，PI将使用支持理论对三角化子范畴进行分类，在代数群具有正特征的无穷小邻域的情况下，寻找复李代数的Beilinson-Bernstein局部化定理的类比，并研究表征理论范畴的Orlov对应。在第四个项目中，这是一个更具有代数性质和几何应用的项目，PI将为一类新的有限维Hopf代数建立上同调的有限生成。