Geometric aspects of representations and cohomology of finite dimensional algebras

Pevtsova建议调查方面的模块表示理论的各种有限维代数领域的积极特征，并表示理论的有关代数领域的特征零。 Pevtsova和其他人的早期工作，采用本地方法在模块化表示理论的基础上，拟议的研究旨在创建新的不变量，并实现更好地理解现有的。例如，Pevtsova建议在联合工作与Friedlander产生更精细的不变量比支持品种考虑限制的陈述子代数同构的群代数的循环群。在另一个项目中，与威瑟斯彭联合，Pevtsova将寻求一个代表性理论描述的支持品种的代表性限制量子李代数在根的单位。在这个项目的一个小姐妹中，Pevtsova和Witherspoon提出了对特征为零的域上的一类局部代数的秩簇进行新的描述，并为某些非上对易的Hopf代数定义了秩簇，将新的秩簇与现有的使用Hochschild上同调的几何结构进行了识别。在第三个项目中，Pevtsova建议建立与李代数的约化包络代数相关的某些三角化模范畴之间的几何对应关系。拟议的研究从理解困难的哲学开始，通过研究嵌入原始结构中的一系列简单结构来理解复杂的代数结构。简单的对象被很好地理解，因此挑战是将这种理解结合起来，以获得关于原始的、困难的结构的一些信息。这个想法是适用于研究正式的代数对象所产生的对称性熟悉的结构。在这个过程中，两个不同的数学领域，几何和代数的相互作用，被广泛使用，揭示了美丽的连接，并使应用程序到这两个领域。这项建议的第二个方面包括指导女大学生，参加一个受高中生欢迎的数学课程，以及编辑一本数学杂志。

Pevtsova proposes to investigate aspects of the modular representation theory of various finite dimensional algebras over a field of positive characteristic, and of the representation theory of related algebras over fields of characteristic zero. Built upon earlier work of Pevtsova and others which employs local methods in modular representation theory, the proposed research aims tocreate new invariants and to achieve a better understanding of existing ones. For example, Pevtsova proposes in joint work with Friedlander to produce finer invariants than support varieties by considering restrictions of representations to subalgebras isomorphic to group algebras of a cyclic group. In another project, joint with Witherspoon, Pevtsova will seek a representation-theoretic description ofsupport varieties for representations of a restricted quantum Lie algebra at roots of unity. In a baby sisterof this project, Pevtsova and Witherspoon propose to develop a new description of the rank variety for a class of local algebras over fields of characteristic zero, and define rank varieties for certain non-cocommutative Hopf algebras, identifying the new varieties with the existing geometric constructions which use Hochschild cohomology. In a third project, Pevtsova proposes to establish a geometric correspondence between certain triangulated module categories associated to reduced enveloping algebras of a Lie algebra.The proposed research begins with the philosophy of understanding a difficult, complicated algebraic structure by studying a family of simple structures embedded in the original. The simple objects are well understood so that the challenge is to combine this understanding to gain some information about the original, difficult structure. This idea is applied to the study of formal algebraic objects arising as symmetries of familiar structures. In the process, the interplay of two different mathematical fields, geometry and algebra, is used extensively, revealing beautiful connections and enabling applications to both areas. A second aspect of this proposal includes the mentoring of female undergraduate students, participation in a popular mathematical program for advanced high school students, and the editing of a special volume of amathematical journal.