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.
