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

这个建议涉及到对方案的动机上同调及其与代数K-理论的关系的研究。该项目的一部分是研究一般可收缩层，研究者试图利用这类层的性质证明Grayson动机上同调与Voevodsky动机上同调是一致的，因此Grayson构造提供了另一种发展动机谱序列的方法。Milnor猜想，希望能为有限广义线性群的同调刚性性质的证明提供新的方法。最后，研究人员计划弄清楚，如果在证明布洛赫-加藤猜想modulo一个任意的素数中缺少什么，这项研究计划是在数学领域被称为代数几何. 代数几何的目标是深入理解多项式方程解的几何性质--当每个人在学校看到二次方程时，更高次或更多变量的方程变得更加微妙。 然而，由于计算机和机器人的计算本质上是对连续的"真实的世界"的有限近似，因此理解多项式对这些努力至关重要。

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.