2011/01/31 by Michael Hutchings · 37 citations
Mathematics · #Advanced Operator Algebra Research #Artificial intelligence #Ball (mathematics) #Combinatorics #Computer science #Disjoint sets #Embedding #Geometric and Algebraic Topology #Geometry #Homotopy and Cohomology in Algebraic Topology #Mathematics #Pure mathematics #Symplectic geometry #Symplectic manifold #math.SG
paper · pdf · doi:10.1073/pnas.1018622108
published in Proceedings of the National Academy of Sciences 108(20), 8093-8099 (National Academy of Sciences) · updated bibliography, corrected typos, to appear in PNAS
arxiv created 2011/02/15 · openalex publication_date 2011/04/25 · arxiv updated 2016/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 result is related to previously known criteria for when a disjoint union of balls can be symplectically embedded into a ball. Numerical invariants defined using embedded contact homology give general obstructions to symplectic embeddings in four dimensions which turn out to be sharp in the above cases.