Deformations and Representation Theory

调查适用于工具从组representationtheory研究变形的陈述profinitegroups和复杂的模块等群体。调查员的主要主题是检查问题和应用程序有关的概括的变形理论的群表示开发的B。马祖尔推广是这个理论的扩展（1）变形的对象在派生类别，（2）变形的表示，其图像位于各种线性代数群，如正交群。调查研究应用这些理论的某些嵌入问题。第二个主题是确定有限群的不可约模表示的泛变形环的环结构，只使用有限群的局部结构。调查员还致力于另一个项目，涉及有限维代数的表示模空间。目标是利用模空间的richalgebro几何结构来更好地理解代数自同构的几何行为。群是抽象的数学对象，人们可以通过它来编码和研究对称性，例如化学分子，晶体，网络或抽象的数学结构。群的表示提供了一种提取群内部结构信息的方法，粗略地说，表示可以被认为是群的“线性快照”，由显式描述的矩阵给出。在这个项目中，研究人员研究变形的表示。一个给定的表象的变形形成了一系列的表象，这些表象以某种方式与这个表象相关联。如果有一个单一的变形可以用来描述所有的变形，我们就说是一个普遍的变形。通用变形提供了通用的结构，它可以用来一次性解决某些问题，否则这些问题将不得不以逐案的方式解决。这是这个项目的目标是研究某些概括的变形的表示，这反过来又会导致更强大的应用程序在不同的领域。

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.