Mathematical Sciences: Localization and periodicity in unstable homotopy theory

作者打算：(1)利用Adams谱序列技巧和Bousfield局部化继续他对不稳定周期同伦群的研究；(2)将与Mark Mahowald关于二次悬挂映射纤维的v2周期同伦群的联合工作推广到更高的周期性；(3)根据同调理论E(N)研究不稳定球的局部化；(4)研究基于E(N)的不稳定Adams-Novikov谱序列：(5)计算自由迭代环空间的mod-p K-理论，并将其应用于v1周期同伦群的计算。拓扑学最初是作为研究微分方程的一种方法出现的。在其间的一个世纪里，它发生了惊人的演变，往往看起来完全有自己的生命，独立于其来源。然而，它已经越来越多地开始偿还债务。问题的关键在于，例如，知道一个方程的解是否为闭合环或是否打结，往往比准确和数值地知道其所有点的坐标更为重要。此外，虽然前者的定性信息隐含在后者的完整描述中，但如何提取它却一点也不明显。由拓扑学家开发的复杂的机器，例如为这个项目概述的同伦理论的研究，现在已经发展到了通常允许以自然的方式处理这些自然问题的程度。

The investigator intends: (1) to continue his study of unstable periodic homotopy groups, using Adams spectral sequence techniques and Bousfield localization; (2) to generalize joint work with Mark Mahowald on v2 periodic homotopy groups of the fiber of the secondary suspension map to higher periodicity; (3) to study the localization of an unstable sphere with respect to the homology theory E(n); (4) to study the unstable Adams-Novikov spectral sequence based on E(n); (5) to compute the mod-p K-theory of free iterated loop spaces and apply this to the calculation of v1 perioidic homotopy groups. Topology arose originally as an approach to the study of differential equations. Over the intervening century it has evolved prodigiously, often seeming to have a life entirely its own and independent of its source. Increasingly, however, it has begun to repay its debt. The point is that it is often more important to know of the solution of an equation whether it is a closed loop or whether it is knotted, for example, than to know precisely and numerically what are the coordinates of all of its points. Furthermore, although the former qualitative information is implicitly contained in the latter complete description, it is not at all obvious how to extract it. The sophisticated machinery that has been developed by topologists, such things as the investigations in homotopy theory outlined for this project, has now been developed to a degree that often permits these natural problems to be handled in a natural way.