Approximating Transcendental Thurston Theory

 A Thurston map is the topological analog of a post-singularly finite (PSF) meromorphic function f that is possibly transcendental. In this talk we will discuss what it means for a Thurston map to be realized as a holomorphic PSF map, and survey known results. Our work in progress attempts to characterize several objects in transcendental planar Thurston theory as limits, in a suitable sense, of corresponding objects from the Thurston polynomial setting. We explore the connection between this research program and the question of holomorphic realization and related problems.