Mathematical Sciences: Motives and Algebraic Cycles

该奖项支持动机和代数循环方面的工作。主要研究者之一将致力于动机上同调；由Gillet和Soule证明的Riemann-Roch型定理的非阿基米德推广；以及岩泽理论对环和动机的推广。另一位主要研究人员将在Grassmannians上使用某些线性构型的族来研究高对数的存在性，并探索Grassmanian复形和地面场的K群之间的关系。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。

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.