Polylogarithms, Moduli Spaces, Mixed Motives and L-Functions

PI想在复曲线的基本群补全上研究真实混合Hodge结构的费曼积分描述。一个特别有趣的例子是泛模曲线，其中费曼积分的相关系数推广了Rankin-Selberg积分。PI想把它们和模形式积的l函数的特殊值联系起来。他想在一般复变的有理同伦型上找到一个与实际混合Hodge结构相似的Feynman积分描述。PI想继续研究曲线的动机基本群及其与模变的关系、经典多对数及其推广、l函数的特殊值、混合动机和动机多重l值。最后，他想继续与V.V. Fock合作研究二维曲面上高Teichm\ uller空间上局部系统的模空间及其量子二对数的量子化，以及与表示理论和三折不变量的关系。在过去的几年里，许多来自物理学的思想对纯数学产生了巨大的影响，反之亦然。到目前为止，数论从这些见解中受益的程度明显低于其他数学领域。PI希望利用物理学家广泛使用的费曼积分、量子二对数和量子变形、量子化和其他工具，研究数论的几个具体问题，以及更广泛的算术代数几何。特别是，他想要证明某些非常特殊的实数，将任意多项式方程系统的一组复解与有理数系数联系起来，并称为有理数上任意变化的有理同伦型的周期，可以定义为费曼积分的相关器。他想在这些数中找到所谓的l函数的特殊值。PI还希望这个与算术代数几何问题相关的费曼积分的具体例子将为物理学中出现的费曼积分的研究带来现代算术代数几何的强大方法。

The PI would like to study a Feynman integral description of the real mixed Hodge structure on a completions of the fundamental group of a complex curve. An especially interesting case is the universal modular curve, where the correlators of the Feynman integral generelise the Rankin-Selberg integrals. The PI wants to relate them to special values of L-functions of products of modular forms. He wants to find a similar Feynman integral description of the real mixed Hodge structure on rational homotopy type of a general complex variety. The PI wants to continue his study of the motivic fundamental groups of curves and their relationship with modular varieties, classical polylogarithms and their generalizations, special values of L-functions, mixed motives and motivic multiple L-values. Finally, he wants to continue his joint work with V.V. Fock on moduli spaces of local systems on 2D-surfaces higher Teichm\"uller spaces, and its quantization using the quantum dilogarithm, and relationship with representation theory and invariants of 3-folds.During the last years many ideas coming from Physics had a tremendous impact on pure Mathematics, and vice versa. Number Theory so far benefited from these insights significantly less then other areas of Mathematics. The PI wants to investigate several concrete problems of Number Theory, and more generally Arithmetic Algebraic Geometry, using Feynman integrls, quantum dilogarithm and quantum deformations, quantisation and other tools widely employed by Physisits. In particular he wants to show that certain very specific real numbers, related the set of complex solutions of an arbitrary system of polynomial equations with rational coefficients, and called periods of the rational homotopy type of an arbitrary variety over rationals, can be defined as correlators of Feynman integrals. He wants to find the so-called special values of L-functions among these numbers. The PI also hopes that this concrete example of a Feynman integral related to an arithmetic algebraic geometry problem will bring powerful methods of mordern arithmetic algebraic geometry to the study of Feynman integrals which appear in Physics.