Lifting and degeneration problems

主要研究人员应用代数表示论和数论中的工具来研究无限群的表示的形变和这种群的模复形。形变和形变环一直是伽罗瓦表示和模形式理论最近显著进展的中心。这个项目有四个主要目标：(1)研究变形与某些嵌入问题解的不存在之间的联系；(2)找到泛形变环的奇点的界；(3)确定有限群表示的泛形变环的环结构；(4)发展复形变的阻塞理论。这位首席研究员还致力于有限维代数上模的退化的另一个项目。第二个项目的主要目标是使用Grassmannians来研究局部模的顶稳简并。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，表示可以被认为是由显式描述的矩阵给出的组的“线性化快照”。在这个项目中，首席研究员研究表示的变形。给定表示的变形形成了以某种方式与该表示相关联的表示族。如果只有一种变形可以用来唯一地描述所有其他的变形，我们就说一种普遍的变形。通用变形提供了通用构造，可用于一次解决某些问题，否则这些问题必须以逐个案例的方式解决。本课题属于表示论和数论的数学领域。过去，这些领域的研究在密码学和纠错码等学科中得到了意想不到的应用，并以这种方式造福于社会。

The principal investigator applies tools from algebraic representation theory and from number theory to study deformations of representations of profinite groups and of complexes of modules for such groups. Deformations and deformation rings have been at the center of remarkable recent progress in the theory of Galois representations and modular forms. This project has four main goals: (1) to study the connection of deformations to the non-existence of solutions to certain embedding problems; (2) to find bounds on the singularities of universal deformation rings; (3) to determine the ring structure of universal deformation rings of representations of finite groups; and (4) to develop an obstruction theory for deformations of complexes. The principal investigator also works on one other project on degenerations of modules over a finite dimensional algebra. The main goal of this second project is to use Grassmannians to study top-stable degenerations of local modules.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 principal 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 uniquely describe all other 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. This project belongs to the mathematical areas of representation theory and number theory. Research in these areas has in the past had unexpected applications to subjects such as cryptography and error correcting codes, and in this way has been a benefit to society.