Mathematical Sciences: Algebraic Cycles and Theory of Motives

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Bloch 9423007 This award continues support of work on the structure of the Chow group of algebraic cycles and related questions in arithmetic and in the theory of motives. The recently constructed spectral sequence from higher Chow groups to algebraic K-theory will be an important tool in developing an integral theory of motives and their extensions. In turn, such an integral theory will provide insight into the arithmetic, and most particularly some geometric understanding of the Bloch-Kato conjecture concerning special values of L-functions, of motives. The principal investigator will also study Arakelov theory for varieties over non-archimedean fields. This should clarify the relation between motives and L-functions when local factors of the L-function have zeroes at the origin. 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 robotics.