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.

主要研究人员适用于工具从代数表示论和数论研究变形的陈述profinite群和复杂的模块等群体。变形和变形环一直是伽罗瓦表示和模形式理论最近取得显著进展的中心。该项目有四个主要目标：（1）研究变形与某些嵌入问题解的不存在性之间的联系;（2）找到通用变形环奇异性的界限;（3）确定有限群表示的通用变形环的环结构;（4）发展复形变形的障碍理论。首席研究员还致力于另一个项目退化的模块在有限维代数。第二个项目的主要目标是利用格拉斯曼群来研究局部模的顶稳定退化。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，网络或抽象的数学结构。组的表示提供了一种提取有关组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群的“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以某种方式与该表示相关联的表示族。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，我们谈论的是一个普遍的变形。通用变形提供了通用的结构，可以用来一次性解决某些问题，否则这些问题将不得不以个案的方式解决。这个项目属于数学领域的表示论和数论。过去，这些领域的研究在密码学和纠错码等学科中有着意想不到的应用，并以这种方式造福于社会。