3d Quantum Trace Map

In 2010, Bonahon and Wong constructed the 2d quantum trace map, connecting two different quantizations of the SL_2(C) character variety of an ideally triangulated hyperbolic surface S. The classical limit of this map expresses the trace of the holonomy of a closed immersed curve in S as a Laurent polynomial in the (square roots of the) shear coordinates for the hyperbolic structure defined with respect to the triangulation. In this talk, we will discuss a new construction of a 3d quantum trace map for ideally triangulated 3-manifolds, focusing on the geometric aspects. This is joint work with Sunghyuk Park.