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

Northwestern University, Evanston IL

Investigators

Abstract

The proposal concerns the investigation of motivic cohomology of schemes and its relationships to algebraic K-theory. One part of the project proposes an investigation of generically contractible sheaves. The investigator intends to use the properties of such sheaves in an attempt to show that Grayson motivic cohomology coincides with Voevodsky one and hence Graysons construction gives another approach 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 proof of the rigidity property for homology of the finite general linear group. Finally the investigator plans to figure out what if anything is missing in the proof of the Bloch-Kato Conjecture modulo an arbitrary prime integer. This research proposal is in the area of mathematics known as algebraic geometry. The objective of algebraic geometry is to gain a deep understanding of the geometric properties of solutions to polynomial equations -- while everyone sees quadratic equations in school, equations of higher degree or more variables become much more subtle. Yet, because computer and robot computations are inherently finite approximations to the continuous "real world," understanding polynomials is vital to such endeavors.