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Thurston's skinning map, together with his Bounded Image Theorem, are a crucial ingredient in his proof of hyperbolization for Haken 3-manifolds. A better quantitative understanding of this map can help us connect topological descriptions of 3-manifolds to their geometric features. Yair Minsky of Yale University discusses some partial results in this direction at the Bill Thurston Legacy Conference, June 23, 2014.
The conference, "What's Next? The mathematical legacy of Bill Thurston," held at Cornell June 23-27, 2014, brought together mathematicians from a broad spectrum of areas to describe recent advances and explore future directions motivated by Thurston's transformative ideas.