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

Goncharov教授继续研究多对数及其推广的算术方面，如多重多对数和量子多对数、代数变体的l函数的特殊值、动力基群、模空间和高量子Teichmuller理论。Goncharov教授研究了射影线的动机基本群的结构，这些射影线在零、无穷和所有n个单位根处穿刺，以及它与模变的几何和拓扑以及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述穿透的投影线的基本群的亲- 1补全的作用。这个故事的分析方面涉及多个zeta值及其推广的性质，在单位的n个根处求值的多个多对数。与模变的几何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局部系统的一些新的模空间及其非交换变形，其中G是一个分裂约化群。这个理论结合了表征理论的许多不同方面当S是简单的但群是一般的，经典的Teichmuller理论对应于G是最简单的情况，也就是2 × 2矩阵的群，而S是一般的。这些模空间的量子化是由动机和量子二对数控制的，从而提供了混合动机和数学物理之间富有成效的关系的一个例子。这项研究是在算术代数几何领域，这是数学的一个分支，涉及整数的问题，但使用来自几何形状研究的思想来接近它们。它是数论的现代版本，数论是一门非常古老的学科，但充满了难题和重要的猜想。整系数多项式方程组理论对于密码学和编码理论等许多应用都具有重要意义。就在最近，物理学的思想开始影响这个主题。申请者将运用数论、代数几何和数学物理的最新技术来研究l函数及其在整数点上的特殊值。

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.