Recent progress on symplectic embedding problems in four dimensions

Abstract

We survey some recent progress on understanding when one four-dimensional symplectic manifold can be symplectically embedded into another. In 2010, McDuff established a number-theoretic criterion for the existence of a symplectic embedding of one four-dimensional ellipsoid into another. This is related to previously known criteria for when a disjoint union of balls can be symplectically embedded into a ball. The new theory of "ECH capacities" gives general obstructions to symplectic embeddings in four dimensions which turn out to be sharp in the above cases.

updated bibliography, corrected typos, to appear in PNAS

References in corpus (4)

• Symplectic embeddings of 4-dimensional ellipsoids: erratum
• New obstructions to symplectic embeddings
• Symplectic embeddings of polydisks
• Embedded contact homology and Seiberg-Witten Floer cohomology I

Cited by in corpus (18)

• Symplectic embeddings into four-dimensional concave toric domains
• Symplectic capacities from positive S^1-equivariant symplectic homology
• Beyond ECH capacities
• Symplectic embeddings of 4-dimensional ellipsoids into cubes
• Ehrhart polynomials and symplectic embeddings of ellipsoids
• Some optimal embeddings of symplectic ellipsoids
• When symplectic topology meets Banach space geometry
• Symplectic embeddings of four-dimensional ellipsoids into integral polydiscs
• The Hofer question on intermediate symplectic capacities
• An elementary alternative to ECH capacities
• The asymptotics of ECH capacities
• On the Ekeland-Hofer symplectic capacities of the real bidisc
• Special eccentricities of rational four-dimensional ellipsoids
• The generating function of the embedding capacity for 4-dimensional symplectic ellipsoids
• Packing numbers of rational ruled 4-manifolds
• Four-periodic infinite staircases for four-dimensional polydisks
• Floer theoretic invariants for 3- and 4-manifolds
• Fourier-Dedekind Sums and an Extension of Rademacher Reciprocity