Polylogarithms, Moduli Spaces, Mixed Motives, and L-Functions

摘要奖DMS-0400449 Goncharov教授Goncharov继续他的研究的算术方面的多面体及其推广，如多重多面体和量子多面体，特殊值的L-函数的代数簇，motivic基本群，模空间和更高的量子Teichmuller理论。Goncharov教授研究了在零，无穷大和所有N阶单位根处穿孔的投影线的motivic基本群的结构，以及它与几何和拓扑的模块化变体和数学物理的惊人关系。问题的算术方面涉及绝对伽罗瓦群对上面所穿孔的射影线的基本群的pro-l完备化的作用。 这个故事的分析方面涉及多重zeta值及其推广的属性，在N次单位根处评估的多重多项式。 模簇与几何的关系以及与数学物理的关系是研究这一问题的新工具。Goncharov教授研究了更高量子Teichmuller理论，该理论研究了曲面S上G-局部系统的一些新模空间，其中G是分裂约化群及其非交换变形。这一理论结合了许多不同方面的代表性理论出现时，S是简单的，但该集团是一般的，和经典的Teichmuller理论对应的情况下，当G是最简单的可能，即集团的两个由两个矩阵，而S是一般的。 这些模空间的量子化是由动机和量子双参数决定的，因此提供了混合动机和数学物理之间富有成果的关系的例子。 这项研究是在算术代数几何领域，这是数学的分支，与整数有关的问题，但使用来自几何形状调查的想法来处理它们。它是数论的现代版本，这是一个非常古老的学科，但充满了困难的问题和重要的成就。整系数多项式方程组的理论对于密码学和编码理论等许多应用都很重要。 就在最近，物理学的思想开始影响这个学科。提出者将利用数论、代数几何和数学物理的最新技术来研究L-函数及其在整数点处的特殊值。

Abstract for award DMS-0400449 of GoncharovProfessor Goncharov continues his study of the arithmetic aspects of polylogarithms and their generalizations, such as multiple polylogarithms and quantum polylogarithms, special values of L-functions of algebraic varieties, motivic fundamental groups, moduli spaces and higher quantum Teichmuller theory. 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 and mathematical physics. The 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. The analytic aspect of the story concerns the properties of multiple zeta values and their generalizations, multiple polylogarithms evaluated at N-th roots of unity. The relationship with the geometry of modular varieties as well as with mathematical physics are new tools to study this problem. Professor Goncharov investigates the higher quantum Teichmuller theory, which studies some new moduli spaces of G-local systems on a surface S, where G is a split reductive group, and its non-commutative deformations. This theory unites many different aspects of the representation theory which appear when S is simple but the group is general, and the classical Teichmuller theory corresponding to the case when G is the simplest possible, that is the group of two by two matrices, while S is general. The quantisation of these moduli spaces is governed by the motivic and quantum dilogarithms, and thus provides an example of fruitful relationship between mixed motives and mathematical physics. This research is in the area of arithmetic algebraic geometry, which is the branch of mathematics that is concerned with questions about the integers, but approaches them using the ideas coming from investigation of geometric shapes. It is a modern version of number theory, which is a very old subject, but is full of difficult problems and significant conjectures. The theory of systems of polynomial equations with integer coefficients is important for many applications including questions in cryptography and coding theory. Just recently, ideas from physics started to influence the subject. The proposer will use the latest techniques in number theory, algebraic geometry and mathematical physics to study the L-functions and their special values at integer points.