Chromatic stable homotopy theory

本文研究球面稳定同伦群的两种方法。 第一个是与日本Hirofumi Nakai合作，扩展了PI的书“复杂的Cobordism和稳定的球面同伦群”的最后一章中描述的方法。“第二个是与弗吉尼亚大学的迈克·希尔和哈佛大学的迈克·霍普金斯的联合工作。 它涉及到研究Morava稳定群的有限子群的同伦不动点集。自从提出这个建议以来，PI和他的两个美国合作者已经使用它（和一个新的）解决了45年前的Arf Kervaire不变问题，这是我们领域最大的问题之一。 在20世纪70年代，有几次不成功的尝试试图解决它，证明有无限多个维度，其中框架流形（某种高维形状）的不变量可以是非零的。 我们的结果是，最多有六个（目前已知五个）这样的维度。 这一说法是如此令人惊讶，以至于现在被称为末日定理。我们开发的技术来证明它是完全不同的，从以前的方法，可以适用于其他问题的拓扑和数学物理。

This research concerns two approaches to the stable homotopy groups of spheres. The first is joint work with Hirofumi Nakai of Japan on extending the methods described in the last Chapter of the PI's book "Complex Cobordism and Stable Homotopy Groups of Spheres." The second is joint work with Mike Hill of the University of Virginia and Mike Hopkins of Harvard University. It involves studying the homotopy fixed point sets of finite subgroups of the Morava stabilizer groups.Since the proposal was written, the PI and his two American collaborators have used the method in it (and a new one) to solve the 45 year old Arf-Kervaire invariant problem, which was one of the biggest questions in our field. In the 1970s there were several unsuccessful attempts to solve it by showing that there are infinitely many dimensions in which the invariant of a framed manifold (a certain kind of higher dimensional shape) can be nonzero. Our result is that there are at most six (five are currently known) such dimensions. This statement is so surprising that it is now known as the Doomsday Theorem. The techniques we developed to prove it are quite different from previous approaches and could be applicable to other problems in topology and mathematical physics.