Mathematical Sciences: Quillen-Type Homology and Homotopy Theory

总的来说，这个项目致力于简化和 使更多的可用几个强大的，复杂的，和微妙的 拓扑学的代数技巧 在20世纪60年代，丹尼尔奎伦 描述了如何定义单纯对象的同调， 许多不同的范畴，包括集合，环上的代数， Steenrod代数上的不稳定代数 许多 同源性理论中的计算可以从这个角度来考虑， 并给出了某些“不稳定亚当斯”的E ~ 2项的计算 谱序列”，例如Bousfield-Kan谱序列。 过去的经验表明了这一办法的力量; 本项目的目的就是深入探讨这一理念。 一 一个特殊的分支将是探索同源性 函数复形的计算方法。 鲍斯菲尔德

In general terms this project is devoted to simplifying and making more usable several powerful, complicated, and subtle algebraic techniques of topology. In the 1960's, Daniel Quillen described how to define the homology of simplical objects over many different categories, including sets, algebras over a ring, and unstable algebras over the Steenrod algebra. Many of the calculations in homology theory can be cast in this light, among them the computation of the E2-term of certain "unstable Adams spectral sequences" such as the Bousfield-Kan spectral sequence. Past experience has indicated the strength of this approach; the purpose of this project is to explore this idea in depth. A particular ramification would be the exploration of the homology of function complexes using tools of Jean Lannes and A.K. Bousfield.