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The recent Abel prize winner Dennis Sullivan switched from Chemical Engineering to majoring in Maths after hearing a talk about an illuminating particular theorem. From here:

The epiphany for me was watching the professor explaining that any surface topologically like a balloon, and no matter what shape - a banana or the statue of David by Michelangelo could be placed on to a perfectly round sphere so that the stretching or squeezing. required at each and every point is the same in all directions at each such point,” he said. Further the correspondence was unique once the location of three points was specified and these points could be specified arbitrarily… “This was general , deep and absolutely beautiful,” he recalls.

What is the exact statement of the theorem, and where can I find a proof?

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This is probably a special case of the uniformization theorem, which (for us) says that any space homeomorphic to a sphere which is equipped with a riemannian metric is conformal to the unit sphere. That is, you're allowed to rescale locally around each point, and the amount by which you rescale varies smoothly.

You can find a discussion about some proofs of this theorem here.


I hope this helps ^_^

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