From symplectic embeddings to number theory II
Responsable du projet Félix Schlenk
Collaborateur Clémence Labrousse
Résumé Symplectic geometry is the geometry underlying Hamiltonian dynamics, and symplectic mappings arise as time 1 maps of Hamiltonian flows. In the last three decades, spectacular rigidity phenomena for symplectic mappings were discovered that show that symplectic mappings are much more special than volume preserving mappings. Recently, McDuff and Schlenk computed the function giving for each r the radius of the smallest 4-dimensional ball into which the 4-dimensional ellipsoid E(1,r) symplectically embeds, and Frenkel and Müller computed the function giving for each r the radius of the smallest 4-dimensional cube into which E(1,r) symplectically embeds. These works show that sometimes, the structure of symplectic rigidity is rich: The graph of both functions is surprisingly complicated. It is given in part in terms of Fibonacci and Pell numbers. Along the proof, Diophantine systems, very special solutions to these systems, and many other number theoretic quantities, relations and identities appear in a rather mysterious way. Around the same time, Hutchings and McDuff have answered the Hofer conjecture, by giving a combinatorial recipe telling whether one 4-dimensional ellipsoid symplectically embeds into another such ellipsoid. The goal of this project is to determine the corresponding embedding functions. In this way, we hope to better understand the structure of symplectic rigidity in an interesting class of examples.
Mots-clés Symplectic geometry
Type de projet Recherche fondamentale
Domaine de recherche Mathématiques
Source de financement FNS - Encouragement de projets (Div. I-III)
Etat Terminé
Début de projet 1-10-2012
Fin du projet 30-9-2015
Budget alloué 302'838.00
Autre information http://p3.snf.ch/projects-144432#
Contact Félix Schlenk