Mathematical Sciences: Homotopy Theory and Its Applications

这个项目包括四位数学家和他们的学生对同伦理论问题的研究。霍普金斯将利用p-进数论和局部化的工具研究稳定同伦理论中的分类问题。Kan将继续研究同伦理论中的基本结构，并结合不稳定同伦理论中的分类问题进行研究。彼得森将把Dickson代数和其他代数作为Steenrod代数上的模来学习。米勒将使用上同调方法研究有限循环空间的结构。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。

This project encompasses research by four mathematicians and their students on problems in homotopy theory. Hopkins will study classification questions in stable homotopy theory using tools from p-adic number theory and localization. Kan will continue research into fundamental structures in homotopy theory connected with classification questions in unstable homotopy theory. Peterson will study the Dickson algebra and other algebras as modules over the Steenrod algebra. Miller will study the structure of finite loop spaces using cohomological methods. 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.