Gromov's non-squeezing theorem | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-30

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Non-squeezing_theorem


00:01:12 1 Background and statement
00:02:55 2 The “symplectic camel”



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SUMMARY
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The non-squeezing theorem, also called Gromov's non-squeezing theorem, is one of the most important theorems in symplectic geometry.  It was first proven in 1985 by Mikhail Gromov. 
The theorem states that one cannot embed a ball into a cylinder via a symplectic map unless the radius of the ball is less than or equal to the radius of the cylinder.  The importance of this theorem is as follows: very little was known about the geometry behind symplectic transformations.
One easy consequence of a transformation being symplectic is that it preserves volume. One can easily embed a ball of any radius into a cylinder of any other radius by a volume-preserving transformation: just picture squeezing the ball into the cylinder (hence, the name non-squeezing theorem). Thus, the non-squeezing theorem tells us that, although symplectic transformations are volume-preserving, it is much more restrictive for a transformation to be symplectic than it is to be volume-preserving.