Mathematical Sciences: Presidential Young Investigator Award

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Hopkins has recently been working out consequences of the nilpotence theorem in stable homotopy theory. The nilpotence theorem determines the prime ideals of the sphere spectrum thought of as a ring up to homotopy. The sphere is analogous to the ring of integers, being an initial object in a suitable category of rings. This analogy becomes more striking when one observes that the "completions" of the sphere at its prime ideals acquire natural compact topologies. He would like to assemble these completions into a global object analogous to the ring of adeles in number theory. Such an object would be appropriate for getting at large-scale aspects of stable homotopy theory. One goal would be to construct L-series with analytic properties measuring the homotopy groups of finite complexes and the homomorphisms induced by maps between them. Fundamental to this is a family of cohomology theories generalizing K-theory, and a basic problem is to represent them geometrically with "generalized vector bundles". A theorem of Atiyah relates the K-theory of the classifying space of a finite group to its character ring. D. Ravenel and Hopkins have recently generalized the notion of character and proved an analogue of Atiyah's result for these exotic K-theories. It is hoped that construction will shed light on what "generalized vector bundles" should be. All of this activity falls under the general heading of exploring and perfecting algebraic techniques for treating qualitative features of spaces, e.g. connectedness, knottedness, stability, and so forth. Such features can be every bit as important for applications as are the usual features examined by methods of classical analysis.