Polylogarithms and Motivic Cohomology

近年来出现了两种主要的动机（共）同源方法。首先，将其构造为若干复的上同调群，这些复的项由显式生成器和推广米尔诺k群的关系给出。第二，可以将一个方案的上同调构造为一个由代数环定义的复的上同调，从而推广了Chow群的经典定义。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。多元对数已经以这样或那样的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有着深刻的联系。这个项目应该揭示这些联系。本研究属于算术代数几何领域，是代数几何与数论技术相结合的学科。在其最初的公式中，代数几何处理的图形可以用最简单的方程，即多项式在平面上定义。数论是研究可以用整数、1、2、3……来表示的数的学科。在上世纪下半叶，这两个看似相距甚远的学科却产生了巨大的相互影响。算术代数几何领域现在使用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.