Modular representation theory, triangulated categories and cohomology

模表示论、三角范畴和上同调

• 批准号: 0800940
• 负责人: Julia Pevtsova
• 金额: $ 8.41万
• 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: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0800940&HistoricalAwards=false
• 关键词: Modular; representation; theory; triangulated; categories

This project is mainly devoted to some aspects of representation theory and cohomology of various families of finite dimensional algebras. The questions range from calculating a specific geometric invariant for particular classes of representations to understanding the (geometric) structure of triangulated categories associated to a given algebraic object. In particular, Pevtsova proposes to continue the study of modules of constant Jordan type and related invariants for a finite group scheme, initiated in a joint work with E. Friedlander and J. Carlson. Pevtsova is seeking further knowledge on cohomology and associated geometric invariants of certain non cocommutative finite dimensional Hopf algebras. The project also aims to investigate the geometric properties of triangulated categories, such as the derived category of perfect complexes of a stack. Bringing the projects on finite group schemes and derived categories to a meeting point, Pevtsova is seeking to compare derived categories associated to different algebraic objects via their geometry.Representation theory as a subject has emerged about 100 hundred years agoin the work of Frobenius and Schur and quickly became an active area of research. In its current stage of development, representation theory has been discovered to be intimately connected to numerous brunches of mathematics, such as geometry, topology and combinatorics, as well as physics. Pevtsova is particularly interested in connections with geometry. Representation theory studies actions of groups and other algebraic structures on vector spaces. In particular, modular representation theory studies actions in a context when they are not semi-simple: not every vector space splits as a direct sums of orbits under the action. Pevtsova studies invariants of such actions which arise from geometric considerations. Her work takes its roots in the fundamental work of Quillen on group cohomology and expands in two different directions: one is to understand and compute invariants for particular actions, the other is to understand global properties of families of vector spaces with an action of a particular group. Pevtsova is also actively involved with math enrichment programs for school children. She will continue running a math challenge program at a local elementary school, and will be teaching at a residential summer math program for high school students from the Northwest organized yearly at the University of Washington.

这个项目主要致力于有限维代数的各种族的表示论和上同调的某些方面。这些问题的范围从计算特定类别的表示的特定几何不变理解（几何）结构的三角形类别与给定的代数对象。特别是，Pevtsova建议继续研究有限群方案的常Jordan型模和相关不变量，这是在与E。Friedlander和J. Carlson。Pevtsova正在寻求进一步的知识上同调和相关的几何不变量的某些非上交换有限维Hopf代数。该项目还旨在研究三角范畴的几何性质，例如堆栈的完全复形的派生范畴。Pevtsova将有限群方案和派生范畴的项目带到了一个交汇点，她正在寻求通过几何比较与不同代数对象相关的派生范畴。表示论作为一门学科出现在大约100年前的Frobenius和Schur的工作中，并迅速成为一个活跃的研究领域。在其目前的发展阶段，表示论已被发现与数学的许多分支密切相关，如几何学，拓扑学和组合学，以及物理学。Pevtsova特别感兴趣的是与几何的连接。表示论研究群和其他代数结构在向量空间上的作用。特别是，模表示论研究的是在非半简单的情况下的作用：不是每个向量空间都分裂为作用下轨道的直和。Pevtsova研究不变量的这种行动所产生的几何考虑。她的工作植根于奎伦的基本工作组上同调和扩大在两个不同的方向：一个是理解和计算不变量的特定行动，另一个是了解全球性质的家庭向量空间的行动一个特定的群体。Pevtsova还积极参与学校儿童的数学丰富计划。她将继续在当地一所小学开展一个数学挑战项目，并将在华盛顿大学每年为西北地区的高中生组织的一个暑期数学项目中任教。