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”支持了马萨诸塞州剑桥市一组拓扑学家的研究。目前的赠款旨在加强和利用由此产生的创造力和专门知识的集中，方法是在夏季召集相当少的在密切相关领域工作的专家小组，并举办每周(或更频繁的)公开研讨会，以传播这项工作并吸引波士顿地区的其他人，特别是研究生。2009年夏天的焦点是Michael Hopkins、Mike Hill和Douglas Ravenel最近对Kervaire不变量问题的解决方案。这个问题的解决出现在几何拓扑学的早期，需要等待半个世纪才能发展出足够强大的同伦理论工具。新的解从本质上发挥了当代稳定同伦理论的全部方法--色、等变和基元。闭合曲面的分类-球面、环面等-在19世纪完成。在20世纪60年代，S描述了一种分类n维流形的策略，它是闭曲面的n维模拟。这个被称为“外科手术”的项目基本上是在50年前完成的，只剩下一个恼人的问题。剩下的问题涉及一个被称为克维埃不变量的微妙的量，当时的方法不足以确定它的价值。代数拓扑学五十年的发展终于解决了这个问题。这笔赠款旨在将参与这项工作的专家聚集在一起，探讨其后果，并在一次研讨会上传播结果，该研讨会将介绍Kervaire不变量问题的背景和意义，然后系统地开发解决该问题所需的技术。

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.