MODULI SPACES, MOTIVES, PERIODS and SCATTERING AMPLITUDES

PI希望利用最近发展的两种数学理论：混合动机理论和簇变理论来研究量子场论中的散射振幅。PI希望进一步发展与格拉斯曼子几何有关的散射振幅的壳上方法，并找到计算散射振幅的有效方法。PI想要研究混合实霍奇轮系派生范畴的费曼积分描述。PI想要研究表示论和几何中的正则基，并将它们与镜像对称联系起来。他想继续研究二维表面上局部系统的模空间及其量化，以及与表征理论和镜像对称的关系。双曲三维流形的理论可以看作是三维流形上的某些局部系统的研究，其值在最简单的复李群之一上。他想为所有复约李群及其量子类比建立一个类似的理论。在过去的几年里，纯数学中的一些新思想对理论物理产生了巨大的影响，反过来，来自物理学的许多思想对纯数学产生了巨大的影响。特别是，数学中最复杂的领域之一——动机理论的思想，在量子场论中散射振幅的计算问题——实验中观察到的数据——中得到了应用。另一方面，量子化的一般思想在纯数学的许多领域得到了具体的实现。PI希望通过使用量子二对数、量子化、量子上同调和费曼积分以及其他受物理学启发的工具来研究数论、代数几何和表示理论的几个具体问题。PI希望在量子场论和代数几何之间寻找新发现的联系，特别是寻找计算散射振幅的有效方法。

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.