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Puncture forgetful maps have been studied for various objects, including Teichmuller spaces, mapping class groups, and closed curves. However, puncture forgetting is known to be discontinuous for curves and the Thurston boundary(=projective measured foliations). In this talk, we will discuss several ideas of forgetting punctures in measured foliations that define upper semi-continuous maps. We then talk about applications to Teichmuller geodesics, the dynamics of post-critically finite rational maps, and point-pushing mapping classes. This is joint work with Jeremy Kahn.