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。它起源于对称性的研究，并在一百多年前的弗罗贝纽斯和舒尔的著作中作为一个独立的学科出现。在其目前的发展阶段，表示论与数学的许多分支密切相关，如几何学，拓扑学，组合学和分析，以及物理学。在这个项目中，PI将建立表征理论和几何之间的新联系，并开发技术，以揭示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。这些举措包括当地小学的数学强化计划，中学生的全市数学圈，公共数学讲座和5-10年级学生的特殊数学奥林匹克。这些机会的目的是吸引更多的年轻人，特别是年轻女性，在数学科学的职业生涯，并提高认识和欣赏美丽的数学性质在下一代...这项研究的重点是四个相互关联的方向可追溯到奎伦的基本工作组上同调从1971年。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，在正特征的代数群的无穷小邻域的情况下找到复李代数的Beilinson-Bernstein局部化定理的类似物，并研究表示论范畴的奥尔洛夫对应。在第四个项目，这是一个更代数性质和几何应用，PI将建立有限代上同调一类新的有限维Hopf代数。