Geometric and Cohomological Invariants in Modular Representation Theory

表示论研究群和其他代数结构在向量空间上的作用。它起源于对对称性的研究，并在一百多年前弗罗贝尼乌斯和舒尔的著作中作为一门学科出现。在目前的发展阶段，表示理论与许多数学分支紧密交织在一起，如几何学、拓扑学、组合学和分析，以及与物理学。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生、初中生和初中生的几项教育推广举措。这些举措包括从当地一所小学的强化数学项目，到面向中学生的全市数学圈子，再到公开数学讲座和为5-10年级学生举办的特别数学奥林匹克竞赛。这些机会旨在吸引更多的年轻人，特别是年轻女性，投身于数学科学的职业生涯，并在下一代提高人们对数学美丽本质的认识和欣赏。这项研究集中在四个相互关联的方向上，这些方向可以追溯到Quillen自1971年以来在群上同调方面的基础工作。该项目的一个重要部分集中在解释与几何术语表示相关联的类别。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，在具有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，并研究表示理论范畴的Orlov对应。在第四个项目中，它具有更多的代数性质和几何应用，PI将建立一类新的有限维Hopf代数的有限上同调生成。

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.