The New York Times, [reporting on Dennis Sullivan's Abel prize](https://www.nytimes.com/2022/03/23/science/abel-prize-mathematics.html), recounts the incident that lured Sullivan from chemical engineering to mathematics:



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>  One day during an advanced calculus lecture, the professor drew two shapes on the blackboard — one a circle, the other more blobby, like a kidney. He then said you could stretch either one to fit on the other.
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> That was not particularly surprising. But then the professor said there was a way — and essentially just one way — to do the stretching such that the stretching was the same in all directions.
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> “This blew my mind,” Dr. Sullivan recalled. “This was not like mathematics I’d learned up to that point. It was much deeper.”
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1. In the first quoted paragraph, what is the meaning of "fit on"?
2. In the second quoted paragraph, what is the meaning of the professor's statement?



Uniqueness of "stretching" (subject to constraints) for a two-dimensional figure

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> In the first quoted paragraph, what is the meaning of "fit on"?
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There is a [homeomorphism](https://en.wikipedia.org/wiki/Homeomorphism) between any pair of (open, Riemannian) disks.



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> In the second quoted paragraph, what is the meaning of the professor's statement?
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There is an (essentially) unique [conformal map](https://en.wikipedia.org/wiki/Conformal_map) from any (open) disk in the plane to the unit disk in the plane. This is (part of) the [uniformisation theorem](https://en.wikipedia.org/wiki/Uniformization_theorem).


