Structured Ring Spectra and Chromatic Stable Homotopy Theory

DMS-0204129保罗·G·戈尔斯同伦理论是研究在相当灵活的连续变形类型下保持不变的拓扑现象。最基本的是，我们试图计算和理解有限单纯复形或CW复形之间的同伦类映射，也许在某种稳定化之后。计算和理解是不一样的：我们有40年或更长时间的计算，但直到最近我们才开始有语言来从我们拥有的数据中建立一个连贯的理论。因此，这个项目既是关于计算的，也是关于探索和改进语言的--尽管两者都将是混合的。切入点是观察到，任何上同调理论和一个好的陈类理论都会产生一个正式的群律，从而产生一个正式的群。相反，给定一个形式群，我们可以问是否存在相应的上同调理论。这些上同调理论描绘了稳定同伦理论的更精细结构。这就是稳定同伦的色图：我们试图利用形式群的代数几何来组织和指导对计算和理论的更深层次结构的研究。这个项目寻求在两个方向上发展这一观点，一个是局部的，一个是全球的。第一个或局部方向是对一般K(N)-局部同伦理论的研究，特别是对K(N)-局部球面的研究。第二个或更具全局性的方向是，使用形式基团的堆栈和模堆作为基本的参数化工具，使我们对结构环谱的知识系统化。

DMS-0204129Paul G. GoerssHomotopy theory is the study of topological phenomena that remain invariant under fairly flexible sorts of continuous deformations. Most basically, we seek to compute and to understand homotopy classes of maps between finite simplicial or CW complexes, perhaps after some sort of stabilization. Computation and understanding are not the same: we have forty or more years of computations, but only lately have we begun to have the language to in which to make a coherent theory out of the data we have. This project is, therefore, as much about exploring and refining the language as it is about calculations -- although there will be a mixture of both. The entry point is the observation that any cohomology theory with a good theory of Chern classes gives rise to a formal group lawand, hence, a formal group. Conversely, given a formal groupwe can ask whether there is an associated cohomology theory.These cohomology theories then delineate the finer structureof stable homotopy theory.This is the chromatic picture of stable homotopy: we seekto use the algebraic geometry of formal groups to organize and direct investigations into the deeper structure of computations and theory. This project seeks to develop this point of view in two directions, one local and one global. The first, or local, direction is an investigation into K(n)-local homotopy theory in general and into the K(n)-local sphere in particular. The second, or more global, direction, would be to make systematicour knowledge of structured ring spectra using stacks and the moduli stack of formal groups as the basic parametrizingdevice.