Polylogarithms, Moduli Spaces, Mixed Motives and L-Functions

PI想研究复曲线基本群的完备化上的真实的混合Hodge结构的Feynman积分描述。一个特别有趣的例子是泛模曲线，其中费曼积分的积分器一般 Rankin-Selberg积分PI希望将它们与模形式乘积的L函数的特殊值联系起来。他希望找到一般复簇的有理同伦型上的真实的混合Hodge结构的类似Feynman积分描述。PI希望继续他的研究motivic基本群的曲线和他们的关系与模品种，经典的多项式和他们的推广，特殊值的L-功能，混合动机和motivic多L-值。最后，他想继续他的联合工作与V. V.福克的模空间的本地系统的二维表面高Teichm\“uller空间，其量化使用的量子双对数，和关系与表示论和不变量的3倍。在过去几年中，许多想法来自物理学产生了巨大的影响，纯数学，反之亦然。到目前为止，数论从这些见解中获益明显少于数学的其他领域。PI希望研究数论的几个具体问题，更普遍的是算术代数几何，使用费曼积分，量子双对数和量子变形，量化和物理学广泛使用的其他工具。特别是他想表明，某些非常具体的真实的数字，有关的一套复杂的解决方案的任意系统的多项式方程的合理系数，并呼吁期间的合理同伦类型的任意品种的理性，可以被定义为principalators的费曼积分。他想在这些数字中找到L函数的所谓特殊值。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.