Motivic Cohomology and K-Theory

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The research project will try to settle old conjectures about the structure of K-theory and Motives, and clarify recently discovered relationships between disparate areas of algebra, algebraic geometry, and algebraic topology. The motive of an algebraic variety is built from geometric relationships, and is designed to be the universal motif for cohomological invariants (hence the name), so its structure reveals aspects of geometry. In contrast, the K-theory of a ring or variety is built from relationships involving linear algebra using algebraic topology, so its structure reveals aspects of algebra and topology. Clarifying the relationships between these constructs will help our understanding of many phenomena that are currently understand only poorly.One part of the project is to find clean proofs of major pieces of the proof of the recently-verified Bloch-Kato conjecture. This includes trying to: find a general degree formula which combines various degree formulas due to Rost, in terms of algebraic cobordism theory; establish an equivalence between models of the symmetric powers of a normal variety; and understand motivic cohomology operations better. These all play a role in the proof. This part of the project will be invaluable in training young researchers around the world. A related part of the project is to understand how Voevodsky's slice filtration is related to algebraic cobordism and other motivic spectra. The goal is to verify several of the slice conjectures, especially in finite characteristic, using homological algebra and algebraic Thom spaces. Part of this will be joint work with graduate students, contributing to their training. The project will also study the relationship between the singularities of a variety, its K-theory and its cdh-cohomology, using recently developed cohomological techniques and the relationship to cyclic homology and de Rham-Witt theory. This should shed some light on classes of singularities related to du Bois singularities. The final part of the project is to formulate an archimedean cohomology for a motive over a number field and relate it to Serre's local factors for the L-functions of the motive. This generalizes work of Connes and Consani, and uses cyclic homology for schemes.