The PI would like to study scattering amplitudes in quantum field theory by using two recently developed mathematical theories: theory of mixed motives and theory of cluster varieties. The PI wants to develop further the on-shell approach to scattering amplitudes related to the geometry of Grassmannians, and to find effective ways to calculate the scattering amplitudes. The PI wants to study a Feynman integral description of the derived category of mixed real Hodge sheaves. The PI wants to study canonical bases in representation theory and geometry, and relate them to the mirror symmetry. He wants to continue his work on moduli spaces of local systems on 2D-surfaces and its quantization, and relationship with representation theory and mirror symmetry. The theory of hyperbolic 3D-manifolds can be viewed as the study of certain local systems on 3D-manifolds with values in one of the simplest complex Lie group. He wants to develop a similar theory for all complex reductive Lie groups, as well as its quantum analog.
During the last years several new ideas in pure mathematics had a big impact on theoretical physics, and vice verse, many ideas coming from physics had a tremendous impact on pure mathematics. In particular, the ideas of one of the most sophisticated areas of mathematics, theory of motives, found its applications in the problem of calculation of scattering amplitudes in quantum field theory - the data observed in experimenters. On the other hand, the general idea of quantization found its concrete realizations in many of areas of pure mathematics. The PI wants to investigate several concrete problems of number theory, algebraic geometry, and representation theory by using quantum dilogarithms, quantization, quantum cohomology and Feynman integrals, and other tools inspired by physics. The PI wants to pursue the newly found links between the quantum field theory and algebraic geometry to find, in particular, effective ways to calculate the scattering amplitudes.
