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.
One of the major outcomes was a close collaboration with a group of theoretical physicicts working on scattering amplitudes, which led to unexpected application of some recently developed mathematical ideas, such as theory of mixed motives and motivic Galois groups, totally positive Grassmannians, cluster coordinates and cluster varieties, to the theory of scattering amplitudes. Another major outcome is application of certain ideas from quantum field theory to the mixed Hodge theory. One of the concrete achievements was a short explicit formula developed jointly with with M. Spradlin, A. Volovich, C.Vergu for the six point two loop scattering amplitude in N=4 superYang-Mills theory given in terms of the classical tetralogarithms, and obtained by using the methods provided by the theory of motives. Another concrete achievment was a new on shell method of calculating scattering Amplitudes in N=4 Super Yang-Mills theory was developed jointly with N. Arkani-Hamed, J.Bourjaily, F.Cachazo, A. Postnikov, J.Trnka. It emphasizes the role of totally positive Grassmannian. In the classical approach scattering amplitudes are calculated as a sum of an enourmous number of integrals related to the Feynman diagrams, and each individual Feynman diagram does not describe a process whioch has physical sense. In the on shell approach each diagram has physical meaning. Jointly with R. Kenyon, we uncovered cluster nature of the dimer model, and described a cluster integrable system related to the dimer model on a torus.
