Mathematical Sciences: Quillen-Type Homology and Homotopy Theory

首席研究员将尝试扩大范围 某些代数技巧的证明几何效用。 在 20世纪60年代的丹尼尔奎伦展示了如何定义同源性， 上同调在许多范畴。 其中包括单纯集， 在那里他获得了拓扑空间的普通上同调， 但是域上的单纯代数的范畴 mod-p Steenrod代数上的单不稳定代数 支持同伦中相关上同调概念 理论 单纯不稳定代数的上同调，例如， 是研究鲍斯菲尔德的E2项的正确背景 Kan谱序列，一个不稳定的Adams型谱序列 由于海恩斯米勒的工作， Jean Lannes和其他人 根据先前的NSF赠款，首席研究员已经 本文成功地研究了各种不同的Quillen型上同调， 代数的范畴 该项目的目的是采取 这些结果，并扩展和应用它们。 他特别 现在应该能够深入研究巴拉特的 解悬谱序列和他的“李环分析器”-两者 30年来一直是猜测和猜测的来源 其他 项目包括结合奎伦型上同调技术 用克里斯秸秆的方法来处理同伦群 一个楔形的两个空间，研究的频谱序列， 计算截面空间的同伦群，以及 探索的同源性的功能复合物使用的工具 Jean Lannes和A.K.鲍斯菲尔德

The principal investigator will attempt to broaden the scope of certain algebraic techniques of proven geometric utility. In the 1960's Daniel Quillen showed how to define homology and cohomology in many categories. These included simplicial sets, where he obtained the ordinary cohomology of topological spaces, but the categories of simplicial algebras over a field and simplicial unstable algebras over the mod-p Steenrod algebra both support an idea of cohomology that is relevant in homotopy theory. Cohomology of simplicial unstable algebras, for example, is the correct context for studying the E2 term of the Bousfield- Kan spectral sequence, an unstable Adams-type spectral sequence which has gained prominence due to the work of Haynes Miller, Jean Lannes and others. Under prior NSF grants, the principal investigator has made a successful study of the Quillen-type cohomology in various categories of algebras. The purpose of this project is to take these results and to extend and apply them. In particular, he should now be able to make an in-depth study of Barratt's desuspension spectral sequence and his "Lie ring analyzer" - both a source of conjecture and speculation for 30 years. Other projects include combining the Quillen-type cohomology techniques with the methods of Chris Stover to approach the homotopy groups of a wedge of two spaces, the study of a spectral sequence for computing the homotopy groups of a space of sections, and an exploration of the homology of function complexes using tools of Jean Lannes and A.K. Bousfield.