Deformations and Representation Theory

University Of Iowa, Iowa City IA

Investigators

Abstract

The investigator applies tools from group representation theory to study deformations of representations of profinite groups and of complexes of modules for such groups. The investigator's main theme is to examine problems and applications related to generalizations of the deformation theory of group representations developed by B. Mazur. The generalizations are extensions of this theory to (1) deformations of objects in derived categories, and (2) deformations of representations whose images lie in various linear algebraic groups, such as orthogonal groups. The investigator studies applications of these theories to certain embedding problems. The second theme is to determine for finite groups the ring structure of the universal deformation ring of an irreducible modular representation, using only the local structure of the finite group. The investigator also works on one other project which concerns moduli spaces of representations of finite dimensional algebras. The goal is to use the rich algebro-geometric structure of moduli spaces to obtain a better understanding of the geometric behavior of algebra automorphisms. Groups are abstract mathematical objects by which one may encode and study symmetry, for example in chemical molecules, crystals, networks, or abstract mathematical structures. Representations of groups provide a way to extract information about the internal structures of a group. Roughly speaking, representations can be thought of as "linearized snapshots" of the group which are given by explicitly described matrices. In this project, the investigator studies deformations of representations. The deformations of a given representation form a family of representations which are associated to this representation in a certain way. In case there is a single deformation which can be used to describe all deformations, one talks about a universal deformation. Universal deformations provide universal constructions which can be used to solve certain problems all at once, which otherwise would have to be solved in a case-by-case fashion. It is the goal of this project to study certain generalizations of deformations of representations which in turn should lead to more powerful applications in different areas.