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.
