Mathematical Sciences: Algebraic Cycles and Theory of Motives

布洛赫9423007奖继续支持关于周氏代数圈群的结构以及算术和动机理论中的相关问题的工作。最近构造的从高阶Chow群到代数K-理论的谱序列将是发展动机及其扩展的整体理论的重要工具。反过来，这样的积分理论将提供对动机的算术的洞察，尤其是对关于L函数的特定值的布洛赫-加藤猜想的一些几何理解。首席研究员还将研究非阿基米德域上变种的Arakelov理论。这理清了当L函数的局部因素在原点为零时，动机与L函数的关系。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。

Bloch 9423007 This award continues support of work on the structure of the Chow group of algebraic cycles and related questions in arithmetic and in the theory of motives. The recently constructed spectral sequence from higher Chow groups to algebraic K-theory will be an important tool in developing an integral theory of motives and their extensions. In turn, such an integral theory will provide insight into the arithmetic, and most particularly some geometric understanding of the Bloch-Kato conjecture concerning special values of L-functions, of motives. The principal investigator will also study Arakelov theory for varieties over non-archimedean fields. This should clarify the relation between motives and L-functions when local factors of the L-function have zeroes at the origin. 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.