Deformations and Representation Theory

作者应用群表示论中的工具研究了有限群的表示的变形和这类群的模的复形。研究人员的主要主题是研究与B.Mazur发展的群表示形变理论的推广有关的问题和应用。推广到(1)派生范畴中对象的变形，以及(2)像位于各种线性代数群中的表示的变形，例如正交群。研究者研究了这些理论在某些嵌入问题上的应用。第二个主题是仅利用有限群的局部结构，确定不可约模表示的泛形变环的环结构。这位研究者还做了另一个项目，它涉及有限维代数表示的模空间。目标是利用模空间的重数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，人们可以用它来编码和研究对称性，例如在化学分子、晶体、网络或抽象的数学结构中。群的表示提供了一种提取关于群的内部结构的信息的方法。粗略地说，群的表示可以被认为是由显式描述的矩阵给出的群的线性化快照。在这个项目中，研究者研究了表象的变形。给定表示的变形形成一族表示，这些表示以某种方式与该表示相关联。如果只有一种变形可以用来描述所有的变形，就可以说是普遍变形。通用变形提供了通用结构，可以用来一次解决某些问题，否则这些问题必须以个案的方式解决。这个项目的目标是研究表示的变形的某些推广，这反过来又应该导致在不同领域中更强大的应用。

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.