K-Theory and Motivic Cohomology

9700881莱文这个项目涉及对理据范畴和其他相关构式的研究。主要研究者将把域上簇的上同调的计算推广到任意维光滑基格式的情况。他计划通过构造一个三角张量范畴来尝试构造光滑射影簇的Chow群上的滤子(沿着Beilinson和Bloch的猜想滤子的路线)；他还将考虑Resnikov关于平面丛的Chern类的零化的结果的相对版本。主要研究人员将Bloch-Gabber-Kato定理推广到系数为p的K-理论在正特征p中的部分计算。他将计算约化群及其分类空间的动机上同调和K-上同调，从而得到关于约化群G的E-G-扭量的新不变量。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25年里，它已经取得了革命性的成就。在它的起源中，它处理的是可以在平面上用最简单的方程定义的图形，即多项式。如今，该领域不仅使用代数的方法，而且使用分析和拓扑学的方法，反过来，在这些领域以及物理、理论计算机科学和机器人中也找到了应用。

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.