Schwarz Lemmas in Discrete Conformal Geometry. 

The Schwarz lemma is a fundamental result in complex analysis, with far-reaching applications across geometric function theory and hyperbolic geometry. After reviewing this classical result, we focus on its discrete counterparts in circle packing theory, originally proposed by Thurston as a model for discrete conformal maps. We further provide geometric interpretations of these discrete Schwarz lemmas as comparison theorems for generalized convex polyhedra in hyperbolic space, and discuss their implications for discrete uniformization theorems of polyhedral surfaces.