Summer Workshop on Homotopy Theory; Cambridge, MA

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市的一组拓扑学家的研究。目前的资助旨在通过在夏季聚集一小群在密切相关领域工作的专家，并举办每周一次（或更频繁）的研讨会，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，来增强和利用由此产生的创造力和专业知识的集中。2009年夏季的焦点是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，在几何拓扑学的早期就出现了，不得不等待半个世纪，直到发展出足够强大的同伦理论工具。新的解决方案从本质上发挥了当代稳定同伦理论的整个范围的方法-色，等变和动机。封闭曲面的分类——球面、环面等等——是在19世纪完成的。在20世纪60年代，人们描述了一种分类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“外科手术”的项目基本上在50年前就完成了，只剩下一个烦人的地方。剩下的这个问题涉及到一个被称为克维尔不变量的微妙量，当时的方法还不足以确定它的值。代数拓扑学发展了50年，现在终于解决了这个问题。本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.