Geometric aspects of representations and cohomology of finite dimensional algebras

Pevtsova建议研究正特征域上各种有限维代数的模表示理论，以及特征零域上相关代数的表示理论。在Pevtsova等人早期工作的基础上，该研究采用了模表示理论中的局部方法，旨在创建新的不变量并更好地理解现有的不变量。例如，Pevtsova在与Friedlander的联合工作中提出，通过考虑对同构于循环群的群代数的子代数的表示的限制，产生比支持簇更精细的不变量。在与威瑟斯彭合作的另一个项目中，Pevtsova将为单位根上的受限量子李代数的表示寻求支持簇的表示论描述。在这个项目的一个小姐妹中，Pevtsova和Witherspoon提出了特征零域上一类局部代数的秩簇的一种新的描述，并定义了某些非上交换的Hopf代数的秩簇，用现有的利用Hochschild上同调的几何结构来识别新的簇。在第三个项目中，Pevtsova建议在与李代数的约化包络代数相关的某些三角化模范畴之间建立几何对应。所提出的研究从通过研究嵌入在原始代数中的一族简单结构来理解困难、复杂的代数结构的哲学开始。简单的对象被很好地理解，因此挑战是结合这种理解来获得关于原始的、困难的结构的一些信息。这一思想被应用于作为常见结构的对称性出现的形式代数对象的研究。在这个过程中，几何和代数这两个不同的数学领域的相互作用得到了广泛的应用，揭示了美丽的联系，并使这两个领域的应用成为可能。这项建议的第二个方面包括指导女本科生，参加一个受欢迎的高中生数学课程，以及编辑一本专门的数学期刊。

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.