W. Thurston's theorems almost all aim to give a purely topological problem an appropriate geometry, or to identify an appropriate obstruction.. We will illustrate this in two examples:
--The Thurston pullback map to make a rational map from a post-critically finite branched cover of the sphere, and
--The skinning lemma, to find a hyperbolic structure for a Haken 3-manifold.
In both cases, either the relevant map on Teichmüller space has a fixed point, solving the geometrization problem, or there is an obstruction consisting a multicurve.[-]

Combination theorems play an important role in several areas of dynamics, geometry, and group theory. In this talk, we will expound a framework to conformally combine Kleinian (reflection) groups and (anti-)holomorphic rational maps in a single dynamical plane. In the anti-holomorphic setting, such hybrid dynamical systems are generated by Schwarz reflection maps arising from univalent rational maps. A crucial technical ingredient of this study is a recently developed David surgery technique that turns hyperbolic conformal dynamical systems to parabolic ones. We will also mention numerous consequences of this theory, including 1. an explicit dynamical connection between various rational Julia and Kleinian limit sets,2. existence of new classes of welding homeomorphisms and conformally removable Julia/limit sets, and3. failure of topological orbit equivalence rigidity for Fuchsian groups acting on the circle.[-]