Discrete uniformization problem for polyhedral surfaces

The classical uniformization theorem for Riemann surfaces is well-established for both compact and non-compact connected surfaces. However,  in the discrete setting, the discrete uniformization theorem is only known for closed polyhedral surfaces.  This talk will propose a discrete uniformization problem for all surfaces and relate it to the classical Weyl problem in the hyperbolic 3-space and the Cauchy-Alexandrov rigidity theorem on convex polytopes.  Some of our progress will be discussed,  including a discrete Schwarz-Pick-Ahlfors lemma and its application.  This is joint work with Yanwen Luo.