Summer Workshop on Homotopy Theory; Cambridge, MA

美国国家科学基金会的合作项目“同伦理论：应用和新维度”支持马萨诸塞州剑桥的一组拓扑学家的研究。本补助金旨在提高和利用由此产生的创造力和专业知识的集中，通过收集一个相当小的专家组在密切相关的领域工作，在夏季，并进行每周一次的公开（或更频繁）研讨会，以传播这项工作，并吸引其他人在波士顿地区，特别是研究生。2009年夏天的焦点是迈克尔霍普金斯，迈克希尔和道格拉斯拉文埃尔最近解决的Kervaire不变量问题。这个问题的解决，出现在早期的几何拓扑，不得不等待一些半世纪的发展足够强大的同伦理论工具。新的解决方案使当代稳定同伦理论的所有方法都发挥了作用--色性、等变和动机。封闭曲面的分类--球面、环面等--在世纪完成。在20世纪60年代，描述了一种用于分类n-流形的策略，n-流形是封闭曲面的n维类似物。这个被称为“手术”的项目在近50年前基本完成，只剩下一个恼人的问题。这个遗留的问题涉及一个被称为Kervaire不变量的微妙数量，当时的方法不足以确定它的值。50年的发展代数拓扑现在终于导致解决这个问题。目前的赠款旨在汇集参与这项工作的专家，探讨其影响，并在研讨会上传播结果，该研讨会将介绍Kervaire不变问题的背景和意义，然后系统地开发解决问题所需的技术。

The collaborative NSF project "Homotopy Theory: Applications and New Dimensions" supports research by a group of topologists based in Cambridge, Massachusetts. The present grant aims to enhance and leverage the resulting concentration of creativity and expertise by gathering a fairly small group of specialists working in closely allied fields, during the summer, and conducting an open weekly (or more frequent) seminar in order to disseminate this work and engage others in the Boston area including especially graduate students. The focus for the summer of 2009 is the recent solution of the Kervaire invariant problem by Michael Hopkins, Mike Hill, and Douglas Ravenel. The resolution of this problem, which arose in the early days of geometric topology, had to wait for some half century for the development of sufficiently powerful homotopy theoretic tools. The new solution brings into play methods from essentially the whole range of contemporary stable homotopy theory -- chromatic, equivariant, and motivic.The classification of closed surfaces - the sphere, the torus, and so on - was accomplished in the nineteenth century. In the 1960's, a strategy was described for classifying n-manifolds, the n-dimensional analogues of closed surfaces. This project, known as "surgery," was essentially completed almost 50 years ago, with one annoying point left over.This remaining question involved a subtle quantity known as the Kervaire invariant, and the methods of the day were not adequate to determine its value. Fifty years of development of algebraic topology has now finally led to a resolution of this question. The present grant aims to bring together experts involved in this work, to explore its ramifications and disseminate the result in a seminar which will present the background and significance of the Kervaire invariant problem and then systematically develop the techniques needed for its resolution.