Modular representation theory, triangulated categories and cohomology

这个项目主要致力于有限维代数族的表示理论和上同调的某些方面的研究。这些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联的三角化类别的(几何)结构。特别是，Pevtsova建议继续研究有限群方案的常Jordan模和相关不变量，这是由E.Friedlander和J.Carlson共同发起的。Pevtsova正在寻求关于某些非余交换有限维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，例如堆叠的完美复形的派生范畴。Pevtsova将有限群方案和派生范畴的研究引向一个交汇点，试图通过它们的几何比较与不同代数对象相关的派生范畴。表示理论作为一门学科在大约100年前出现在Frobenius和Schur的工作中，并迅速成为一个活跃的研究领域。在目前的发展阶段，表示理论已被发现与许多数学分支密切相关，如几何学、拓扑学、组合学以及物理学。佩夫索娃对几何学的联系特别感兴趣。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，模表示理论在一个不是半简单的上下文中研究动作：并不是每个向量空间都作为动作下的轨道的直接和来分裂。Pevtsova研究由几何因素引起的这些动作的不变量。她的工作植根于Quillen关于群上同调的基本工作，并在两个不同的方向上展开：一个是理解和计算特定行动的不变量，另一个是了解具有特定群行动的向量空间族的全局性质。Pevtsova还积极参与为学龄儿童提供数学丰富的项目。她将继续在当地的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的高中生举办的寄宿暑期数学项目任教。

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.