Representations of 3-manifold groups

Fock和Goncharov定义了李群中曲面群的坐标表示。对于线性群，这些坐标已经被PI和他的合作者推广到3-流形群的表示。坐标允许精确计算表示和有效计算不变量，如体积和陈-西蒙斯不变量。该计划将研究这些坐标的性质，维护和开发计算数据库，研究在量子拓扑学中的应用，并推广到其他李群和高维流形。在解决问题时，用正确的坐标表示它是非常重要的。一个问题可能在一组坐标中完全难以处理，而在另一组坐标中却有一个优雅的解决方案。PI和他的合作者已经定义了特殊的坐标表示的3流形组，这是重要的对象，研究三维空间。这些坐标允许以前难以处理的计算，并且计算揭示了新的和有趣的现象。该提案将研究这些坐标的性质，在许多方向推广，维护和开发计算数据库，并研究量子拓扑学和数学物理学中的应用。

Fock and Goncharov have defined coordinates for representations of surface groups into a Lie group. For linear groups, these coordinates have been generalized by the PI and his collaborators to representations of 3-manifold groups. The coordinates allow for exact computations of representations and efficient computation of invariants such as volume and Chern-Simons invariant. The proposal will study the properties of these coordinates, maintain and develop databases of computations, study applications in quantum topology, and generalize to other Lie groups and higher dimensional manifolds.When solving a problem it is very important to express it in the right coordinates. A problem can be completely intractable in one set of coordinates while having an elegant solution in another. The PI and his collaborators have defined special coordinates for representations of 3-manifold groups, which are important objects for studying 3-dimensional spaces. These coordinates allow for computations that were previously intractable, and the computations have revealed new and interesting phenomena. The proposal will study the properties of these coordinates, generalize in many directions, maintain and develop databases of computations, and study applications in quantum topology and mathematical physics.