Character Varieties and Cluster Mutations

每一个曲面都可以近似为一个由三角形粘合而成的空间。例如，在计算机图形学中，动画图形由移动的多面体表示。类似地，在爱因斯坦相对论中很重要的三维弯曲空间，可以用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胶合模式的具体组合学来研究空间的抽象属性。这种方法非常适合计算机实验，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。研究者和他的学生将探索新的结构，推导新的不变量公式，进行广泛的计算机实验，并将基础理论扩展到更高的维度。该项目将研究Fock和Goncharov引入的旗帜的某些配置空间。这些位形空间有具体的坐标和非常有趣的结构。坐标产生明确的坐标较高的Teichmuller空间的三角曲面，并坐标（装饰）字符品种的三角形3流形。簇结构的坐标也产生了明确的公式不变量，如陈-西蒙斯不变量。这项工作将调查的配置空间的结构，并探讨他们的关系，多面体。 该项目旨在推导出新的不变量公式，并将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.