From symplectic embeddings to number theory
Responsable du projet | Félix Schlenk |
Collaborateur |
David Egbert Frenkel
Dorothee Cosima Mueller Yochay Jerby |
Résumé |
Symplectic geometry is the geometry underlying Hamiltonian dynamics,
and symplectic mappings arise as time-1-maps of Hamiltonian flows.
In the last two decades, spectacular rigidity phenomena for
symplectic mappings were discovered, that show that symplectic
mappings are much more special than volume preserving mappings. At
the same time, various methods were found to construct certain
symplectic mappings. Despite this progress, it is still quite
mysterious what a symplectic mapping is. Recently, in joint work
with Dusa McDuff, we have calculated the function c(r) computing
the radius of the smallest 4-dimensional ball into which the
4-dimensional ellipsoid E(1,r) symplectically embeds. The graph of
this function is surprisingly rich, and is given in part in terms
of Fibonacci numbers. Along the proof, many other number theoretic
quantities, relations and identities appear in a rather mysterious
way. We have also shown that for this problem, the invariant
obtained by the Embedded Contact Homology of Hutchings and Taubes
is a complete invariant. The goal of this project is to compute the
corresponding function for similar symplectic embedding problems,
such as embedding ellipsoids into ellipsoids and polydiscs. In this
way, we hope to better understand these number theoretic aspects of
the answer, and to see how strong the invariant from Embedded
Contact Homology is in other problems. |
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 | 1-10-2010 |
Fin du projet | 30-9-2012 |
Budget alloué | 222'708.00 |
Autre information |
http://p3.snf.ch/projects-132000# |
Contact | Félix Schlenk |