Algebraic Methods in Stable Homotopy Theory

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DMS-0404651Douglas C. RavenelThis proposal has two parts: Higher chromatic analogs of elliptic cohomology and finding more stable homotopy groups of spheres. The first part concerns a recent discovery by Ravenel that the Jacobians of certain algebraic curves in characteristic P have interesting 1-dimensional formal groups as formal summands. More precisely, he has examples where the the height can be any multiple of p-1. This could lead to analogs of elliptic cohomology, which takes us deeper into the chromatic tower, meaning beyond v2-periodic phenomena. It is the first example known to topologists of formal group laws of height greater than two occuring in a geometric setting. The methods used here come from algebraic geometry and number theory. The second part concerns the determination of the stable homotopy groups of spheres and the Adams-Novikov spectral sequence. This problem has been one of the central and most difficult in algebraic topology since the groups were defined in 1935. Little progress has been made in this area since the the publication of Ravenel's 1986 book (republished in 2003) "Complex Cobordism and the Stable Homotopy Groups of Spheres" which described the state of the art at the time. In it he determined the stable homotopy groups of spheres through dimension 108 for p=3 and 999 for p=5. The latter computation was a substantial improvement over prior knowledge, and neither has been improved upon since. It is generally agreed among homotopy theorists that it is not worthwhile to try to improve our knowledge of stable homotopy groups by a few stems, but that the prospect of increasing the know range by a factor of p would be worth pursuing. This possibility may be within reach now, due to a better understanding of the previously used methods of and improved computer technology.This project will help the general advance of algebraic topology, a subject which has been central to pure mathematics since its founding by Poincare a century ago. It has been a continuing source of new ideas in algebraic geometry as seen in the work of Lefschetz in the '30s, the efforts leading the proof of the Weil conjectures in the'60s and '70s, and most recently in the successful application of motivic cohomology to the Milnor conjecture by Voevodsky. It has also found numerous applications in differential geometry and theoretical physics. The University of Rochester is one of the leading centers of algebraic topology in the world.