Mathematical Sciences: Motives and Algebraic Cycles

该奖项支持动机和代数循环方面的工作。 其中一位主要研究人员将致力于动机上同调； Gillet 和 Soule 证明的黎曼-罗赫型定理的非阿基米德推广；以及岩泽理论对周期和动机的概括。 另一位主要研究者将利用格拉斯曼阶上某些线性配置族来研究更高对数的存在性，并探索格拉斯曼复合体与地场 K 群之间的关系。 这是代数几何领域的研究，代数几何是现代数学最古老的部分之一，但在过去四分之一个世纪中取得了革命性的繁荣。 在它的起源中，它处理可以通过最简单的方程（即多项式）在平面上定义的图形。 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，它正在这些领域以及理论计算机科学和机器人技术中找到应用。

This award supports work on motives and algebraic cycles. One of the principal investigators will work on motivic cohomology; a non-archimedean generalization of the Riemann-Roch type theorem proved by Gillet and Soule; and a generalization of Iwasawa theory to cycles and motives. The other principal investigator will work on the existence of higher logarithms using families of certain linear configurations over the Grassmannians, as well as explore the relationship between the Grassmannian complex and the K-groups of the ground field. 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.