K-Theory and Motivic Cohomology

9876729这个提议涉及到动机上同调的各个方面的研究，以及它与K-理论和霍奇理论的关系。 其中一个项目是将Bloch的高阶Chow群的理论推广到混合特征的方案，并在这样的方案上构造从高阶Chow群到相干层范畴的K-理论的谱序列。 该项目的另一个目标是建立高级Chow群的预期性质，为规则方案提供良好的动机上同调理论，以及上述谱序列的预期性质。 第二个项目是给出一个完整的版本的贝林森-Lichtenbaum的结构使用全纯版本的motivic上同调作为替代的模块化理论。 第三个项目是研究具有对数极点和积分周期的微分形式的一些模性质，第四个项目是给出经典群的积分动机的计算。 数论的历史根源在于对自然数的研究。 它是最古老的数学分支之一。 在过去的半个世纪，它已成为一个不可或缺的工具，在不同的应用领域，如数据传输和处理，通信系统。

9876729The proposal involves a study of various aspects of motivic cohomology and its relationship with K-theory and Hodge theory. One project is to extend the theory of Bloch's higher Chow groups to schemes in mixed characteristic and to construct a spectral sequence from the higher Chow groups to the K-theory of the category of coherent sheaves on such a scheme. A further goal of this project is to establish the expected properties of the higher Chow groups to furnish a good theory of motivic cohomology for regular schemes, as well as the expected properties of the above-mentioned spectral sequence. A second project is to give an integral version of the Beilinson-Lichtenbaum conjectures using a holomorphic version of motivic cohomology as a replacement for the modular theory. A third project is to investigate some modular properties of differential forms with log poles and integral periods, and the fourth project is to give a computation of the integral motive for classical groups.This research is in the area of number theory. Number theory has its historical roots in the study of natural numbers. It is among the oldest branches of mathematics. Within the last half century it has become an indispensible tool in diverse applications in areas such as data transmission and processing, and communication systems.