9700881 Levine This project is concerned with the study of motivic categories and other related constructions. The principal investigator will extend the computation of motivic cohomology of varieties over a field to the case of a smooth base scheme of arbitrary dimension. He plans to attempt a construction of a filtration on the Chow groups of smooth projective varieties (along the lines of the conjectural filtration of Beilinson and Bloch) by constructing a triangulated tensor category satisfying a universal property with respect to motivic cohomology and Hodge cohomology; he also will consider a relative version of the results of Resnikov on the vanishing of Chern classes of flat bundles. The principal investigator will extend the Bloch-Gabber-Kato theorem to a partial computation of K-theory with mod p coefficients in positive characteristic p. He will compute the motivic cohomology and K-cohomology of reductive groups and their classifying spaces, giving rise to new invariants for etale G-torsors for reductive groups G. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.
