Algebraic K-theory, Motivic Cohomology and Homology of Linear Groups

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The proposal concerns the investigation of motivic cohomology ofschemes and its relationships to algebraic K-theory. One part of theproject proposes an investigation of generically contractible sheaves.The investigator intends to use the properties of such sheaves inan attempt to show that Grayson motivic cohomology coincides with Voevodsky one and hence Graysons construction gives anotherapproach to the dvelopment of the motivic spectral sequence.Another part of the project concerns the Friedlander-Milnor Conjecture.The investigator expects to develop new methods for the proofof the rigidity property for homology of the finite generallinear group. Finally the investigator plans to figure outwhat if anything is missing in the proof of the Bloch-Kato Conjecturemodulo an arbitrary prime integer.This research proposal is in the area of mathematics known asalgebraic geometry. The objective of algebraic geometry is togain a deep understanding of the geometric properties of solutions to polynomial equations -- while everyone sees quadratic equations in school, equations of higher degree ormore variables become much more subtle. Yet, becausecomputer and robot computations are inherently finiteapproximations to the continuous "real world," understandingpolynomials is vital to such endeavors.