Problems in Homotopy Theory and Group Cohomology

同伦论和群上同调中的问题

• 批准号: 0604206
• 负责人: Nicholas Kuhn
• 金额: $ 16万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-15 至 2010-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604206&HistoricalAwards=false
• 关键词: Problems; Homotopy; Theory; Group; Cohomology

Professor Kuhn has a long record of developing homotopy and representation theoretic methods to solve problems in both topology and algebra. Recently, he has discovered some remarkable connections between two of the main strands of homotopy theory as studied over the past 25 years: homotopy as organized into its `periodic' layers (as in the work of M.Hopkins), and homotopy as organized using sophisticated homotopical algebraic decomposition techniques (as in the work of T.Goodwillie). The largest part of this project concerns the continued exploration of these connections, and the continued development of tools for exploiting this. A second part will focus on some problems in finite group cohomology exploiting primitives associated to central extensions. Specific older work of his to be brought to bear on these questions include his work on the Whitehead Conjecture, Bousfield--Kuhn telescopic functors, Hopkins--Kuhn--Ravenel generalized characters, and unstable modules over the Steenrod algebra.Homotopy theory and group cohomology are mathematical subjects in which one is trying to discover, and ultimately classify, fundamental "building blocks" of structure: homotopy dealing with deformations of geometric objects such as higher dimensional surfaces, and group cohomology being concerned with analyzing symmetries of both geometric and algebraic objects. A sample homotopy problem is to understand the "shape" of all continuous functions from a surface to itself. A sample group cohomology problem is to understand the way in which very basic subgroups of symmetries "control" the behavior of more complicated symmetry groups. Professor Kuhn is studying these subjects by developing a variety of state-of-the-art algebraic and homotopy theoretic tools. By their very nature, many of these tools involve "universal" constructions of interest to researchers in other scientific disciplines ranging from computer science to robotics.

库恩教授有一个长期的记录发展同伦和表示理论的方法来解决问题的拓扑和代数。 最近，他发现了过去25年来研究的同伦理论的两个主要分支之间的一些显着联系：同伦被组织成它的“周期性”层（如M.霍普金斯的工作），同伦被组织成使用复杂的同伦代数分解技术（如T.古德威利的工作）。 该项目的最大部分涉及继续探索这些联系，并继续开发利用这些联系的工具。 第二部分将集中在有限群上同调利用与中心扩展相关的原语的一些问题。具体的老工作，他将承担这些问题包括他的工作怀特海猜想，Bousfield-库恩望远镜函子，霍普金斯-库恩-拉文埃尔广义字符，和不稳定的模块的Steenrod代数。同伦理论和群上同调是数学科目中的一个试图发现，并最终分类，基本的“积木”结构：同伦处理变形的几何对象，如高维表面，和群上同调正在关注分析对称性的几何和代数对象。 同伦问题的一个例子是理解从曲面到其自身的所有连续函数的“形状”。一个样本群上同调问题是为了理解对称性的基本子群“控制”更复杂对称群的行为的方式。库恩教授正在通过开发各种最先进的代数和同伦理论工具来研究这些主题。就其本质而言，这些工具中的许多工具涉及从计算机科学到机器人技术等其他科学学科的研究人员感兴趣的“通用”结构。