Theorems in geometry | Symplectic geometry

Non-squeezing theorem

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 theorem is important because formerly very little was known about the geometry behind symplectic maps. 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. (Wikipedia).

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From playlist Multivariable Calculus

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Squeeze Theorem

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From playlist Calc 1

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From playlist New Calculus Video Playlist

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From playlist Calculus I

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From playlist Calculus

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From playlist Sequences

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From playlist Mathematics

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From playlist Mathematics

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From playlist MIT 18.100A Real Analysis, Fall 2020

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Alberto Abbondandolo University of Pisa, Italy February 8, 2012 I will discuss a middle-dimensional generalization of Gromov's Non-Squeezing Theorem. For more videos, visit http://video.ias.edu

From playlist Mathematics

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From playlist Mathematics

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From playlist Calculus 1 Exercises

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From playlist MIT 18.100A Real Analysis, Fall 2020

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From playlist Limits

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From playlist Mathematics

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From playlist Mathematics

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From playlist Calculus

Related pages

Phase space | Uncertainty principle | Liouville's theorem (Hamiltonian) | Canonical coordinates | Symplectic geometry | Volume | Canonical transformation | Squeeze mapping