Symplectic embeddings of 4-dimensional ellipsoids

As has been known since the time of Gromov’s Nonsqueezing Theorem, symplectic embedding questions lie at the heart of symplectic geometry. This talk will 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 exceptional curves in blow ups of the complex projective plane. For Michael Atiyah on his 80th birthday The standard symplectic structure on R 2n is: ω0 = dx1 ∧ dx2 + dx3 ∧ dx4 + · · · + dx2n−1 ∧ dx2n. By Darboux’s theorem every symplectic form is locally diffeomorphic to this one, so it is crucial to understand its properties. Gromov’s question: What are the images of a symplectic ball? How are the symplectic and volume preserving cases different? Let B: = B2n √ (a) be the standard (closed) ball of radius a (so a is proportional to a 2-dimensional area) and let φ: B s ↩ → R2n be a symplectic embedding (i.e. a smooth embedding onto φ(B) that preserves ω0). The problem is to describe φ(B) – for example: can it be long and thin? Since ω n 0 = n! dx1 ∧ · · · ∧ dx2n is a volume form, every symplectic embedding preserves volume.