The odd primary analogue to the Milnor conjecture is called the ``Bloch-Kato'' conjecture. Its resolution would permit the proof of the odd primary Quillen-Lichtenbaum conjecture for number fields, and Voevodsky has outlined a program to prove it as well, by analogy with his methods for the Milnor conjecture. What remains to be done is the construction of varieties with appropriate topological behavior, within the A1-theory. M. Rost has been working on these constructions, and the it appears likely that the proof of this conjecture is now within range.

Recent work of G. Carlsson has proposed another approach to the study of the K-theory of fields, more along the lines of the standard descent arguments which motivated the Quillen-Lichtenbaum conjectures originally. The key object in the standard descent argument is the so called ``homotopy fixed point set'' of the action of the absolute Galois group G of a field F on the algebraic closure of F. In terms of the modern language of ring spectra and module spectra, one can view this action as a module structure on the K- theory spectrum of the algebraic closure of F over the KF group ring of G, where KF denotes the K-theory spectrum of F, a ring spectrum. Carlsson's new approach involves the introduction of a larger ring spectrum of operators, including the KF-group ring, which act on the K-theory of the algebraic closure of F. Descent definitely works over this ring spectrum of operators, and the question becomes how to analyze it. Carlsson proposes a model for this ring of operators in terms of the homological algebra of the representation ring of the group G. If the model is correct, it will provide a good homotopy theoretic model for the K-theory of a field, not only the K-groups. The model appears to be correct for a number of fields whose absolute Galois group is topologically cyclic. If this model is correct, it would shed additional light on the relationship of the K-theory of a field with constructions over the Galois group.

We propose to bring Voevodsky and Rost to the American Institute of Mathematics in Palo Alto for the months of April and May of 1999. There would be two goals. The first is the completion of the proof of the Bloch-Kato conjecture by Voevodsky and Rost, and the second would be to develop an understanding of the relationship between the A1 and motivic methods of Voevodsky with the modified descent procedure of Carlsson. The Bloch-Kato conjecture has been outstanding for some time, and is one of the most important opoen problems in the subject. Understanding the relationship between motivic theory and Carlsson's modified descent procedure appears likely to shed a great deal of light on the structure of the K-theory of fields, particularly since the methods start from such completely different points of view. Algebraic K-theory is currently moving ahead at a very fast clip. It would be very valuable to bring together these workers in the area.

Voevodsky and Rost would be working at the American Institute of Mathematics (AIM) during their stay. They would have access to Stanford facilities, such as libraries and computers. Carlsson would spend at least three days per week at AIM. During that time, there would be intense discussions about the problems proposed for the project. If progress warrants it, there would also be some seminar type lectures, open to the mathematical community at large.

- The results predicted by the two approaches agree for free and
free abelian profinite
absolute Galois groups. That is, the homotopy groups of the model spectrum for
algebraic K-theory constructed by Carlsson agree with the homotopy
groups coming
out of the motivic approach for these absolute Galois groups.
- There is a map of spectral sequences from the Carlsson descent
spectral sequence to
the Bloch Lichtenbaum spectral sequence, which should be an isomorphism of
spectral sequences if Carlsson's "ascent conjecture" is valid.
- The Bloch-Kato conjecture can be interpreted as an isomorphism
between certain
derived functors over the representation ring of the absolute Galois group with
cohomology groups of this group.
- Carlsson's ascent conjecture holds for free and free abelian absolute Galois groups.