Mathematical Sciences: Research in Algebra and Algebraic Number Theory

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This project is concerned with research on motives and motivic cohomology. The principal investigator will continue his research in the various properties of his new definitions of motives and notivic cohomology. Given any quasiprojective scheme X over an algebraically closed field k, the Suslin complex of X should give the mixed motive of X as an object in the derived category of abelian groups. The principal investigator has already shown that this is a very natural definition for X an arbitrary curve over k, and he will investigate the case of surfaces. In addition, he will investigate the extent to which his generalized definition of the Suslin complex can be shown to satisfy the desired axioms. This research is concerned with algebraic K-theory. In a broad sense algebraic K-theory concerns th 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.