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.
