Polylogarithms, moduli spaces, Hodge theory, motives and L-functions

PI想研究复杂簇有理同伦型上实混合Hodge结构的费曼积分描述。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概括了 Rankin-Selberg 积分。 PI 希望将它们与模块化形式的乘积的 L 函数的特殊值联系起来。他想要找到混合实霍奇滑轮的派生范畴的费曼积分描述。 PI 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、混合动机和动机多重 L 值的关系。最后，他想继续与 V.V. 合作。关注二维表面更高 Teichmuller 空间上局部系统的模空间及其使用量子二对数的量化，以及与表示论超卡勒几何和 3 倍不变量的关系。在过去的几年中，许多来自物理学的想法对纯数学产生了巨大影响，反之亦然。到目前为止，数论从这些见解中受益的程度远远少于其他数学领域。 PI 希望使用费曼积分、量子二对数和量子变形、量子化以及物理学家广泛使用的其他工具来研究数论以及更一般的算术代数几何的几个具体问题。特别是，他想证明某些非常具体的实数，与具有有理系数的任意多项式方程组的复解集相关，以及有理数上的任意簇的所谓有理同伦类型的周期，可以定义为费曼积分的相关器。他想在这些数字中找到所谓的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 the rational homotopy type of complex varieties. An especially interesting case is the universal modular curve, where the correlators of the Feynman integral generelize 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 Feynman integral description of the derived category of mixed real Hodge sheaves. 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 Teichmuller spaces and its quantization using the quantum dilogarithm, and relationship with representation theory hyperkahler geometry 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 has so far benefitted 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 integrals, quantum dilogarithms and quantum deformations, quantization and other tools widely employed by physicists. In particular he wants to show that certain very specific real numbers, related to the set of complex solutions of an arbitrary system of polynomial equations with rational coefficients, and so-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 modern arithmetic algebraic geometry to the study of Feynman integrals which appear in physics.