Problems in Homotopy Theory and Group Cohomology

库恩教授长期致力于发展同伦和表示理论方法来解决拓扑和代数问题。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著联系：同伦被组织成它的“周期”层（如霍普金斯的工作），同伦被组织成复杂的同伦代数分解技术（如古德威利的工作）。该项目的最大部分是继续探索这些联系，并继续开发利用这些联系的工具。第二部分将重点讨论利用与中心扩展相关的原语的有限群上同调中的一些问题。他对这些问题的具体研究包括怀特海猜想、Bousfield- Kuhn望远镜函子、Hopkins- Kuhn- Ravenel广义特征和Steenrod代数上的不稳定模。同伦理论和群上同调是试图发现并最终分类结构的基本“构建块”的数学主题：同伦处理几何对象的变形，如高维曲面，群上同调涉及分析几何和代数对象的对称性。一个样本同伦问题是理解从一个曲面到它自身的所有连续函数的“形状”。一个样本群上同问题是为了理解非常基本的对称子群如何“控制”更复杂的对称群的行为。库恩教授通过开发各种最先进的代数和同伦理论工具来研究这些课题。就其本质而言，这些工具中的许多都涉及其他科学学科（从计算机科学到机器人技术）研究人员感兴趣的“通用”结构。

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.