Lifting and degeneration problems

提升和退化问题

• 批准号: 0651332
• 负责人: Frauke Bleher
• 金额: $ 9.46万
• 依托单位: University of Iowa
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-15 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0651332&HistoricalAwards=false
• 关键词: Lifting; degeneration; problems

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.

首席研究员应用代数表示论和数论的工具来研究有限群和此类群的模复形表示的变形。变形和变形环一直是伽罗瓦表示和模形式理论最近取得的显着进展的中心。该项目有四个主要目标：（1）研究变形与某些嵌入问题不存在解的关系； (2) 求通用变形环奇点的界限； (3)确定有限群表示的通用变形环的环结构； (4) 发展复合体变形的阻碍理论。首席研究员还致力于另一个关于有限维代数上模的退化的项目。第二个项目的主要目标是利用格拉斯曼尼亚人来研究局部模块的顶稳定退化。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、网络或抽象数学结构中的对称性。群的表示提供了一种提取有关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明确描述的矩阵给出的组的“线性化快照”。在这个项目中，主要研究者研究表征的变形。给定表示的变形形成一系列表示，这些表示以某种方式与该表示相关联。如果有一个单一的变形可以用来唯一地描述所有其他变形，我们就可以谈论通用变形。通用变形提供了通用结构，可用于一次性解决某些问题，否则这些问题必须以具体情况具体分析的方式解决。该项目属于表示论和数论的数学领域。这些领域的研究过去在密码学和纠错码等学科中取得了意想不到的应用，并以这种方式造福了社会。