K-Theory and Motivic Cohomology

9700881 Levine这个项目是关于动机范畴和其他相关结构的研究。主要研究者将域上变量的动机上同调的计算推广到任意维的光滑基格式的情况。他计划通过构造一个满足动机上同调和Hodge上同调的全称性质的三角化张量范畴，尝试在光滑射影变的Chow群上构造一个过滤（沿着Beilinson和Bloch的猜想过滤的路线）；他还将考虑Resnikov关于平束Chern类消失的结果的一个相对版本。主要研究者将把Bloch-Gabber-Kato定理推广到在正特征p中具有mod p系数的k理论的部分计算。他将计算约化群及其分类空间的动机上同调和k上同调，从而得到约化群g -环量g的新不变量。这是代数几何领域的研究。代数几何是现代数学中最古老的部分之一，但在过去的四分之一个世纪里，它已经有了革命性的发展。在它的起源中，它处理的图形可以用最简单的方程，即多项式，在平面上定义。如今，该领域不仅使用代数的方法，还使用分析和拓扑的方法，相反，它在这些领域以及物理学、理论计算机科学和机器人技术中也得到了应用。

9700881 Levine This project is concerned with the study of motivic categories and other related constructions. The principal investigator will extend the computation of motivic cohomology of varieties over a field to the case of a smooth base scheme of arbitrary dimension. He plans to attempt a construction of a filtration on the Chow groups of smooth projective varieties (along the lines of the conjectural filtration of Beilinson and Bloch) by constructing a triangulated tensor category satisfying a universal property with respect to motivic cohomology and Hodge cohomology; he also will consider a relative version of the results of Resnikov on the vanishing of Chern classes of flat bundles. The principal investigator will extend the Bloch-Gabber-Kato theorem to a partial computation of K-theory with mod p coefficients in positive characteristic p. He will compute the motivic cohomology and K-cohomology of reductive groups and their classifying spaces, giving rise to new invariants for etale G-torsors for reductive groups G. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.