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.
