MODULI SPACES, MOTIVES, PERIODS and SCATTERING AMPLITUDES

PI希望使用最近发展起来的两个数学理论来研究量子场论中的散射幅度：混合动机理论和团簇变种理论。PI希望进一步发展与Grassmannians几何有关的散射幅度的壳上方法，并找到计算散射幅度的有效方法。PI想要研究混合实Hodge滑轮的派生范畴的Feynman积分刻划。PI希望学习表示理论和几何中的正则基，并将它们与镜像对称性联系起来。他想继续研究2D曲面上局部系统的模空间及其量子化，以及与表象理论和镜像对称性的关系。双曲三维流形理论可以看作是对三维流形上的某些局部系统的研究，这些流形的值属于最简单的复李群之一。他想为所有的复约化李群及其量子类比发展一个类似的理论。在过去的几年里，纯数学中的几个新想法对理论物理产生了很大的影响，反过来，许多来自物理学的想法对纯数学产生了巨大的影响。特别是，动机理论是数学中最复杂的领域之一，它的思想在计算量子场论中的散射幅度--在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找到了它的具体实现。PI希望利用量子双对数、量子化、量子上同调和费曼积分等受物理学启发的工具来研究数论、代数几何和表象理论的几个具体问题。PI想要追寻新发现的量子场论和代数几何之间的联系，以找到特别是计算散射幅度的有效方法。

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.