The investigator intends: (1) to continue his study of unstable periodic homotopy groups, using Adams spectral sequence techniques and Bousfield localization; (2) to generalize joint work with Mark Mahowald on v2 periodic homotopy groups of the fiber of the secondary suspension map to higher periodicity; (3) to study the localization of an unstable sphere with respect to the homology theory E(n); (4) to study the unstable Adams-Novikov spectral sequence based on E(n); (5) to compute the mod-p K-theory of free iterated loop spaces and apply this to the calculation of v1 perioidic homotopy groups. Topology arose originally as an approach to the study of differential equations. Over the intervening century it has evolved prodigiously, often seeming to have a life entirely its own and independent of its source. Increasingly, however, it has begun to repay its debt. The point is that it is often more important to know of the solution of an equation whether it is a closed loop or whether it is knotted, for example, than to know precisely and numerically what are the coordinates of all of its points. Furthermore, although the former qualitative information is implicitly contained in the latter complete description, it is not at all obvious how to extract it. The sophisticated machinery that has been developed by topologists, such things as the investigations in homotopy theory outlined for this project, has now been developed to a degree that often permits these natural problems to be handled in a natural way.
