, Volume 4, Issue 2, pp 121-139
Date: 25 Dec 2009

Symplectic embeddings and continued fractions: a survey

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Abstract.

As has been known since the time of Gromov’s Non-squeezing Theorem, symplectic embedding questions lie at the heart of symplectic geometry. After surveying some of the most important ways of measuring the size of a symplectic set, these notes discuss some recent developments concerning the question of when a 4-dimensional ellipsoid can be symplectically embedded in a ball. This problem turns out to have unexpected relations to the properties of continued fractions and of exceptional curves in blow ups of the complex projective plane. It is also related to questions of lattice packing of planar triangles.

This article is based on the 6th Takagi Lectures that the author delivered at the Hokkaido University on June 6 and 7, 2009. A related course of lectures was also given at the MSRI Graduate Summer school in August 2009.
Dusa McDuff: partially supported by NSF grant DMS 0604769.
Communicated by: Kaoru Ono