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

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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.