In 1980s, Thurston’s formulated the geometrization conjecture for 3-manifolds, and proved the hyperbolization theorem. The keys to Thurston’s proof are two bounded results for certain deformation spaces of Kleinian groups. In early 1990s, motivated by Thurston’s boundedness theorem and the Sullivan dictionary, McMullen conjectured that certain hyperbolic components of rational maps are bounded.
In this talk, I will start with a historical introduction, and discuss a general strategy of the proof of Thurston’s boundedness theorem. I will then explain how a similar strategy could work for rational maps, and discuss some recent breakthrough towards McMullen's boundedness conjecture.