Mathematical Sciences: Algebraic Geometry and Motivic Cohomology

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Professor Bloch will be working on Motives in algebraic geometry, and their cohomology. He has introduced a complex of higher algebraic cycles for the purpose of calculating the extension groups in the abelian category of "mixed" motives, and plans to study this complex and how it may be employed as a vehicle for realizing the category of mixed motives as a subcategory of the derived category of abelian groups. This is research in the field of Algebraic Geometry, one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origins, 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 theoretical Computer Science and in Robotics.