Polylogarithms, Moduli Spaces, Mixed Motives and L-Functions

PI想要研究复曲线基本群完备化的实混合Hodge结构的Feynman积分描述。一个特别有趣的例子是通用模曲线，其中费曼积分的相关器推广了Rankin-Selberg积分。PI想要将它们与L的特殊值-模形式乘积的函数联系起来。他想找到一般复簇的有理同伦型上的实混合Hodge结构的类似Feynman积分刻画。作者希望继续研究曲线的动机基本群及其与模簇、经典多对数及其推广、L函数的特殊值、混合动机和动机重L值之间的关系。最后，他希望继续与V.V.Fock合作，研究2D曲面高阶Teichm\“Uller空间上局部系统的模空间，以及它使用量子双对数的量子化，以及与表示理论和三重不变量的关系。在过去的几年里，来自物理学的许多想法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远远低于数学的其他领域。PI想要研究数论的几个具体问题，以及更广泛的算术代数几何，使用费曼积分、量子双对数和量子变形、量子化和物理广泛使用的其他工具。具体地说，他想证明某些非常具体的实数可以定义为Feynman积分的相关器，这些实数与具有有理系数的任意多项式方程组的复解集有关，称为有理数上任意变化的有理同伦型周期。他想在这些数字中找到L函数的所谓特殊价值。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把现代算术代数几何的强大方法引入到物理学中出现的费曼积分的研究中。

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.