Mathematical Sciences: Algebraic K-Theory and Motivic Cohomology

该奖项支持方案的动机上同调及其与代数K-理论的关系的研究。将代数K-理论与高阶Chow群联系起来的谱序列的最新进展，以及充分高余维复变簇的高阶Chow群的计算等方面的发展，使人们对有限系数小于或等于2的光滑复变簇的代数K-理论有了完整的理解。主要研究者将继续这方面的研究，以获得对高维复变簇的代数K-理论和更一般方案的理解。除了这方面的研究外，他还将研究多相对代数K-理论与循环同调之间的关系，从而完成了一系列关于代数K-理论消去性的工作。这是对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，在这些领域以及理论计算机科学和机器人学中也找到了应用。

This award supports the study of motivic cohomology of schemes and its relationships to algebraic K-theory. Recent developments concerning a spectral sequence relating algebraic K-theory to higher Chow groups and the computation of the higher Chow groups of complex varieties of sufficiently high codimension in terms of etale cohomology give a complete understanding of algebraic K-theory with finite coefficients of smooth complex varieties of dimension less than or equal to 2. The principal investigator will pursue further this line of investigation in order to obtain an understanding of algebraic K-theory of higher dimensional complex varieties and of more general schemes. In addition to this study of motivic cohomology, he will investigate the relationship between polyrelative algebraic K-theory and cyclic homology, thus completing a series of works on excision in algebraic K-theory. This is research in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origins, 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 theoretical computer science and robotics.