Polylogarithms and Motivic Cohomology

近年来出现了两种主要的理据(上)同调方法。首先，试图将其构造为某些复形的上同调群，其中的项由显式生成元给出，关系推广了Milnor的K-群。其次，可以将方案的上同调构造为用代数圈定义的复数的上同调，从而推广了经典的Chow群定义。每种方法都有自己的优点和缺点。赵建强的研究将继续他的工作，通过使用多对数及其推广来理解这两种方法。多对数以一种或另一种形式存在了几个世纪，但直到最近才有了完整的理论。它们与数论和现代数学物理都有着深厚的联系。这个项目应该会阐明这些联系。这项研究是在算术代数几何领域进行的，这是一门结合了代数几何和数论技术的学科。在最初的公式中，代数几何处理的是可以在平面上由最简单的方程定义的图形，即多项式。数论是研究可以由整数、1、2、3表示的数…在上个世纪后半叶，这两个看似相距遥远的学科产生了巨大的影响。算术代数几何领域现在使用了所有现代数学的技术。

There are two main approches to motivic (co-)homology appearing inrecent years. First, one tries to construct it as cohomology groupsof certain complexes with terms being given by explicitgenerators and relations generalizing Milnor's K-groups. Second, onecan construct cohomology of a scheme as cohomology of a complexdefined in terms of algebraic cycles thus generalizing the classicaldefintion of Chow groups. Each has its own advantages and disadvantages.Dr. Jianqiang Zhao's research will continue his workon understanding both approaches by usingpolylogarithms and their generalizations. Polylogarithms have been around for centuries in one form oranother, but only recently has their whole theory become available. They have deep connections with both number theory and modern mathematical physics. This project should shed light on these connections. This research is in the field of arithmetic algebraic geometry, a subject that combines the techniques of algebraic geometry and numbertheory. In its original formulation, algebraic geometry treated figures that could be defined in the plane by the simplest equations, namely polynomials. Number theoryis the study of numbers that can be expressed interms of whole numbers, 1, 2, 3... In the second halfof last century these two seemingly far apart subjects have produced tremendous impact on each other. The field of arithmetic algebraic geometry now uses techniques from all of modern mathematics.