Character Varieties and Cluster Mutations

每个曲面都可以由一个由三角形粘合而成的空间来近似。例如，这一事实被用于计算机图形学中，其中动画图形由移动的多面体来表示。同样，在爱因斯坦的相对论中很重要的三维弯曲空间，可以用将四面体粘合在一起得到的空间来表示。这个项目将通过使用粘合模式的具体组合来研究空间的抽象性质。这种方法非常适合于计算机实验，参与研究的学生可以在不完全理解抽象理论的情况下进行实验。研究人员和他的学生将探索新的结构，推导不变量的新公式，进行广泛的计算机实验，并将基本理论扩展到更高的维度。该项目将研究Fock和Goncharov引入的某些旗帜的配置空间。这些配置空间有具体的坐标和非常有趣的结构。这些坐标给出了三角化曲面的高阶TeichMuller空间上的显式坐标，以及三角化三维流形的(装饰)特征变量上的坐标。坐标的团簇结构也给出了不变量的显式公式，如Chern-Simons不变量。这项工作将研究配置空间的结构，并探索它们与多对数的关系。该项目旨在推导出不变量的新公式，并将该理论扩展到更高的维度。

Every surface can be approximated by a space obtained by gluing together triangles. This fact is used, for example, in computer graphics where animated figures are represented by moving polyhedra. Similarly, curved spaces in three dimensions, which are important in Einstein's relativity theory, can be represented by spaces obtained by gluing together tetrahedra. This project will study abstract properties of a space by using the concrete combinatorics of the gluing pattern. This approach is well suited for computer experimentation, and students participating in the research can perform experiments without fully understanding the abstract theory. The investigator and his students will explore new structures, derive new formulas for invariants, perform extensive computer experiments, and extend the underlying theory to higher dimensions.The project will study certain configuration spaces of flags introduced by Fock and Goncharov. These configuration spaces have concrete coordinates and very interesting structures. The coordinates give rise to explicit coordinates on higher Teichmuller spaces for triangulated surfaces, and to coordinates on the (decorated) character variety of triangulated 3-manifolds. The cluster structure of the coordinates also gives rise to explicit formulas for invariants such as the Chern-Simons invariant. The work will investigate the structure of the configuration spaces and explore their relationship to polylogarithms. The project aims to derive new formulas for invariants and extend the theory to higher dimensions.