Polylogarithms, Mixed Motives and Special Values of L-Functions

Goncharov教授继续研究经典多重数及其推广的算术方面，如重多重数、代数簇的L函数的特殊值、代数K理论和Motivic Galois群。Goncharov教授研究了射影直线的动机基本群的结构，以及它与模簇的几何和拓扑之间的惊人关系。动机基本群体是一种混合动机。混合动机可以通过他们的霍奇和L的实现来考察。问题的L-进，即算术方面涉及绝对伽罗华群在L前完成如上射影直线的基本群上的作用。当N为1时，这是Grothendieck、Deligne、Ihara、Drinfeld和许多其他数学家研究的经典问题。对于一般的N，这个问题的最简单情况等价于经典的割圆单位理论。模簇与几何的关系是研究这一问题的新工具。这个故事的霍奇，也就是分析方面涉及到多个Zeta值及其推广的性质，即在单位根N处计算的多个多对数。Goncharov教授研究了一个类似的问题，即复乘法在扭点处被穿孔的椭圆曲线的模基本群的结构及其与模簇几何的关系。贡查罗夫教授继续研究L函数和多项式的特殊值。这项研究是在数论领域进行的，数论是数学的一个分支，研究整系数多项式方程的整数和根问题。整系数多项式方程组理论在密码学和编码学中有着重要的应用。这种方程组的一个基本不变量是它的L函数。在过去的三百年里，L函数是新数学概念和理论的主要来源之一。例如，某些L函数提供了链中的重要环节，导致了最近费马大定理的证明。本文利用数论和代数几何的最新方法，研究了L函数及其在整点上的特殊值。

Professor Goncharov continues his study of the arithmetic aspects of classical polylogarithms and their generalizations, such as multiple polylogarithms, special values of L-functions of algebraic varieties, algebraic K-theory and motivic Galois groups. Professor Goncharov investigates the structure of the motivic fundamental group of the projective line punctured at zero, infinity and all N-th roots of unity and its surprising relationship with the geometry and topology of modular varieties. The motivic fundamental group is a mixed motive. Mixed motives can be investigated via their Hodge and l-adic realizations. The l-adic, i.e. arithmetic, side of the problem concerns the action of the absolute Galois group on the pro-l completion of the fundamental group of the projective line punctured as above. When N is 1 it is a classical problem studied by Grothendieck, Deligne, Ihara, Drinfeld and many other mathematicians. The simplest case of this problem for general N is equivalent to the classical theory of cyclotomic units. The relationship with the geometry of modular varieties is a new tool to study this problem. The Hodge, i.e. analytic, aspect of the story concerns the properties of multiple zeta values and their generalizations, multiple polylogarithms evaluated at N-th roots of unity. Professor Goncharov investigates a similar problem about the structure of the motivic fundamental group of an elliptic curve with complex multiplication punctured at the torsion points and its relationship with the geometry of modular varieties. Professor Goncharov continues his study of special values of L-functions and polylogarithms.This research is in the area of number theory, which is the branch of mathematics that is concerned with questions about the integers and roots of polynomial equations with integer coefficients. The theory of systems of polynomial equations with integer coefficients is important for many applications including questions in cryptography and coding theory. A fundamental invariant of such a system of equations is its L-function. During the last three hundred years the L-functions were one of the main sources of new mathematical conceptions and theories. For example, certain L-functions provided vital links in the chain that led to the recent proof of Fermat's last theorem. The proposer uses the latest techniques in number theory and algebraic geometry to study L-functions and their special values at integer points.