Representations of 3-manifold groups

Fock和Goncharov定义了表面群表示为Lie群的坐标。对于线性群，PI和他的合作者已将这些坐标推广到3-流形群的表示。坐标允许精确的表示计算和有效的不变量计算，例如体积和Chern-Simons不变量。该方案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，并将其推广到其他李群和高维流形。一个问题在一组坐标下可能是完全难以处理的，而在另一组坐标下却有一个完美的解决方案。PI和他的合作者为三维流形群的表示定义了特殊的坐标，这是研究三维空间的重要对象。这些坐标允许进行以前难以处理的计算，这些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，向多个方向推广，维护和开发计算数据库，并研究在量子拓扑和数学物理中的应用。

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.