Deformations and Representation Theory

点击翻译按钮获取中文摘要

The investigator applies tools from group representationtheory to study deformations of representations of profinitegroups and of complexes of modules for such groups. Theinvestigator's main theme is to examine problems andapplications related to generalizations of the deformationtheory of group representations developed by B. Mazur. Thegeneralizations are extensions of this theory to (1)deformations of objects in derived categories, and (2)deformations of representations whose images lie in variouslinear algebraic groups, such as orthogonal groups. Theinvestigator studies applications of these theories to certainembedding problems. The second theme is to determine for finitegroups the ring structure of the universal deformation ring ofan irreducible modular representation, using only the localstructure of the finite group. The investigator also works onone other project which concerns moduli spaces of representationsof finite dimensional algebras. The goal is to use the richalgebro-geometric structure of moduli spaces to obtain a betterunderstanding of the geometric behavior of algebra automorphisms.Groups are abstract mathematical objects by which one may encodeand study symmetry, for example in chemical molecules, crystals, networks,or abstract mathematical structures. Representations of groups providea way to extract information about the internal structures of a group.Roughly speaking, representations can be thought of as "linearizedsnapshots" of the group which are given by explicitly describedmatrices. In this project, the investigator studies deformations ofrepresentations. The deformations of a given representation form afamily of representations which are associated to this representationin a certain way. In case there is a single deformation which can beused to describe all deformations, one talks about a universaldeformation. Universal deformations provide universal constructions whichcan be used to solve certain problems all at once, which otherwise wouldhave to be solved in a case-by-case fashion. It is the goal of thisproject to study certain generalizations of deformations ofrepresentations which in turn should lead to more powerful applicationsin different areas.