Mathematical Sciences: K-Theory & Motivic Cohomology

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Levine The principal investigator will continue his study of the motivic category and other related constructions. He will investigate the relationship between the integral motivic cohomology and Bloch's higher Chow groups, to construct a weight structure and a duality on the motivic category, and to relate the motivic Tate category with the motivic Lie algebra constructed by Bloch and Kriz. He plans to lift various constructions in Hodge theory to the motivic setting and to study mod n motivic cohomology. The principal investigator will also use some new ideas of Hoobler to attack Kato's conjecture and will give a construction of a candidate for weight two cohomology. This research is concerned with algebraic K-theory. In a broad sense algebraic K-theory concerns the evolution of concepts from linear algebra such as basis and vector space. This work has significant implications for number theory and algebraic geometry, and promises to make exciting connections between a number of different areas in mathematics. ***