K-Theory and Motivic Cohomology

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9876729The proposal involves a study of various aspects of motivic cohomology and its relationship with K-theory and Hodge theory. One project is to extend the theory of Bloch's higher Chow groups to schemes in mixed characteristic and to construct a spectral sequence from the higher Chow groups to the K-theory of the category of coherent sheaves on such a scheme. A further goal of this project is to establish the expected properties of the higher Chow groups to furnish a good theory of motivic cohomology for regular schemes, as well as the expected properties of the above-mentioned spectral sequence. A second project is to give an integral version of the Beilinson-Lichtenbaum conjectures using a holomorphic version of motivic cohomology as a replacement for the modular theory. A third project is to investigate some modular properties of differential forms with log poles and integral periods, and the fourth project is to give a computation of the integral motive for classical groups.This research is in the area of number theory. Number theory has its historical roots in the study of natural numbers. It is among the oldest branches of mathematics. Within the last half century it has become an indispensible tool in diverse applications in areas such as data transmission and processing, and communication systems.