Mathematical Sciences: Algebraic Geometry and Motivic Cohomology

布洛赫教授将致力于研究代数几何中的动机及其上同调。为了计算“混合”动机的阿贝尔范畴中的扩张群，他引入了一个具有更高代数圈的复合体，并计划研究这个复合体以及它如何被用作实现混合动机范畴作为阿贝尔群派生范畴的一个子范畴的工具。这是对代数几何领域的研究，它是现代数学中最古老的部分之一，但在过去的25年里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，这个领域不仅使用了代数的方法，而且使用了分析和拓扑学的方法，相反，它在这些领域以及理论计算机科学和机器人学中都得到了应用。

Professor Bloch will be working on Motives in algebraic geometry, and their cohomology. He has introduced a complex of higher algebraic cycles for the purpose of calculating the extension groups in the abelian category of "mixed" motives, and plans to study this complex and how it may be employed as a vehicle for realizing the category of mixed motives as a subcategory of the derived category of abelian groups. 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 in Robotics.