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.
