New Developments in Algebraic K-Theory

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Project Abstract for "New Developments in Algebraic K-Theory"This project concerns the algebraic K-theory of fields. In the last two years, V. Voevodsky has revolutionized algebraic K-theory with the methods he uses in his proof of the Milnor conjecture. The methods he has developed suggest a program which will be developed over the next few years to understand algebraic K-theory through a so-called stable homotopy category of algebraic varieties. Using his methods, it has been possible to verify the Quillen Lichtenbaum conjecture in some cases, notably for number fields at the prime two. Part of this project is to continue this work, some joint with M. Rost, to prove this conjecture in a larger number of cases. G. Carlsson has developed a revised version of the Quillen Lichtenbaum conjecture, which should explicitly identify all homotopy groups of the algebraic K-theory spectra, not only the high dimensional ones as in the original conjecture. This version involves the representation theory of the absolute Galois group, rather than just the Galoios cohomology. The second goal of this project is to understand the relationship between The two constructions, in the hopes that Voevodsky's method might prove the revised conjecture, or that Carlsson's method will shed light on the general relationship between Voevodsky's work and the Quilen Lichtenbaum conjecture.Algebraic K-theory is a relatively new subject in mathematics, which promises to relate two seemingly very disparate areas, namely topology (the study of curves and surfaces and their deformations), and the very classical area of number theory, which studies the properties of the integers and equations over the integers. Algebraic K-theory, which is defined topologically, appears to carry with it a great deal of arithmetic information. Two important conjectures concerning this relationship are the Quillen Lichtenbaum conjecture and the Bloch-Kato conjecture. V. Voevodsky has revolutionized this area over the last two or three years with his breakthrough work on both conjectures. This project will be devoted to developing this work further, by having Voevodsky and Rost continue their work on the Bloch-Kato conjecture, and by connecting Voevodsky's methods with a revised, stronger version of the Quillen Lichtenbaum conjecture proposed by Carlsson. In the long run, solutions to these problems have the potential to help us study the sets of solutions in the integers of algebraic equations, such as the famous Fermat equation.