Ehrhart functions and symplectic embeddings of ellipsoids

2013/07/21 by Dan Cristofaro‐Gardiner, Daniel Cristofaro-Gardiner, Aaron Kleinman · 20 citations
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Citation #Computer science #Library science #Nonlinear Waves and Solitons #math.CO #math.SG #msc:11P21 #msc:53D05

paper · pdf · doi:10.1112/jlms.12299

published in Journal of the London Mathematical Society 101(3), 1090-1111 (Wiley) · 24 pages, with 1 figure

arxiv created 2013/07/21 · openalex created_date 2016/06/24 · openalex publication_date 2020/01/27 · arxiv updated 2020/02/05 · openalex updated_date 2026/08/05

Abstract

McDuff has previously shown that one four-dimensional symplectic ellipsoid can be symplectically embedded into another if and only if a certain combinatorial criteria holds. We reinterpret this combinatorial criteria using the theory of Ehrhart quasipolynomials, and we use this to give purely combinatorial proofs of theorems of McDuff–Schlenk and Frenkel–Müller, concerning the existence of ‘infinite staircases’ in symplectic embedding problems. We then find a third, new, staircase and conjecture that these are the only three staircases for embeddings into rational ellipsoids. Several other applications are also discussed; for example, we give new examples of triangles whose Ehrhart function exhibits a period collapse.

Citations

• Maximal periods of (Ehrhart) quasi-polynomials
• Pseudo holomorphic curves in symplectic manifolds
• Embedded contact homology and Seiberg–Witten Floer cohomology I
• Recent progress on symplectic embedding problems in four dimensions
• Symplectic embeddings of 4-dimensional ellipsoids into cubes
• Quasi-period collapse and GL_n(Z)-scissors congruence in rational polytopes
• The ghost stairs stabilize to sharp symplectic embedding obstructions