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

研究了方案的动机上同调及其与代数k理论的关系。该项目的一部分提出了一般可收缩轴的调查。研究者打算利用这些序列的性质来证明Grayson动机上同与Voevodsky动机上同是一致的，因此Grayson结构为动机谱序列的发展提供了另一种方法。该项目的另一部分涉及弗里德兰德-米尔诺猜想。研究者期望开发新的方法来证明有限一般线性群的同调的刚性性质。最后，研究者计划找出布洛赫-加藤猜想（Bloch-Kato猜想）对任意素数的证明中是否遗漏了什么。这项研究计划是在数学领域称为代数几何。代数几何的目标是深入理解多项式方程解的几何性质——虽然每个人在学校都看到二次方程，但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“真实世界”的有限近似，因此理解多项式对这种努力至关重要。

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.