Abstract: The analogy between Ratner's theorems on unipotent flows on homogeneous spaces and the SL(2,R)-action on moduli space has been used profitably, with seminal work of Eskin-Mirzakhani achieving a classification of invariant and stationary measures for this action. Meanwhile, recent investigation into the horocycle flow on moduli space has revealed that this dynamical system exhibits behaviors distinct from unipotent flows on Lie group quotients by lattices. We report on recent work suggestive of a different interpretation of the analogy between moduli space and homogeneous space, and speculate on a number-theoretic origin of the parallels that emerge.