Mathematical Sciences: Homotopy Theory and Its Applications

点击翻译按钮获取中文摘要

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.