2017/02/13 by Dan Cristofaro-Gardiner, Daniel Cristofaro‐Gardiner, Richard Hind +1
Materials Science · Mathematics · #Advanced Combinatorial Mathematics #Context (archaeology) #Ellipsoid #Embedding #Fano plane #Fibonacci number #Geometric and Algebraic Topology #Lattice (music) #Quasicrystal Structures and Properties #Symplectic geometry #math.SG #msc:53D05
paper · pdf · doi:10.1112/topo.12055
79 pages, 1 figure
arxiv created 2017/02/13 · openalex created_date 2017/06/05 · openalex publication_date 2018/04/11 · arxiv updated 2018/04/18 · openalex updated_date 2026/08/06
In determining when a four-dimensional ellipsoid can be symplectically embedded into a ball, McDuff and Schlenk found an infinite sequence of ‘ghost’ obstructions that generate an infinite ‘ghost staircase’ determined by the even index Fibonacci numbers. The ghost obstructions are not visible for the four-dimensional embedding problem because strictly stronger obstructions also exist. We show that in contrast, the embedding constraints associated to the ghost obstructions are sharp for the stabilized problem; moreover, the corresponding optimal embeddings are given by symplectic folding. The proof introduces several techniques of independent interest, namely: (i) new applications of ideas from embedded contact homology (ECH) in the context of nodal curves, (ii) an improved version of the index inequality familiar from the theory of embedded contact homology, (iii) new applications of relative intersection theory in the context of neck stretching analysis, (iv) a new approach to estimating the ECH grading of multiply covered elliptic orbits in terms of areas and continued fractions, and (v) a new technique for understanding the ECH of ellipsoids by constructing explicit bijections between certain sets of lattice points.