Modular representation theory, triangulated categories and cohomology

本课题主要研究有限维代数各种族的表示理论和上同调的一些方面。问题的范围从计算特定表示类的特定几何不变量到理解与给定代数对象相关的三角化类别的（几何）结构。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，该研究是与E. Friedlander和J. Carlson共同发起的。Pevtsova正在寻求关于某些非协易有限维Hopf代数的上同调和相关几何不变量的进一步知识。该项目还旨在研究三角化范畴的几何性质，例如堆栈的完美复合体的派生范畴。Pevtsova将有限群方案和衍生类别的项目带到一个交汇点，试图通过几何来比较与不同代数对象相关的衍生类别。表征理论作为一门学科在大约100年前Frobenius和Schur的工作中出现，并迅速成为一个活跃的研究领域。在目前的发展阶段，表征理论已被发现与许多数学分支密切相关，如几何、拓扑学和组合学，以及物理学。佩夫佐夫对几何学的联系特别感兴趣。表示理论研究群和其他代数结构在向量空间上的作用。特别地，模表示理论研究的是在非半简单的情况下的动作：不是每个向量空间都分裂为动作下轨道的直接和。Pevtsova研究了这些由几何因素引起的不变量。她的工作植根于Quillen关于群上同调的基础工作，并向两个不同的方向扩展：一个是理解和计算特定动作的不变量，另一个是理解具有特定群动作的向量空间族的整体性质。佩夫索娃还积极参与为学龄儿童开设的数学强化课程。她将继续在当地一所小学开展数学挑战项目，并将在华盛顿大学每年为西北地区高中生举办的暑期数学寄宿项目中授课。

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.