Diffeomorphisms preserving a symplectic form enjoy many rigidity properties. One of the most striking, called the Arnold conjecture, is that the number of fixed points of those generated by Hamiltonian functions is bounded from below by the topology of the manifold. This question was translated by Sheila Sandon to contact geometry in terms of translated points of contactomorphisms. Together with G. Granja, Y. Karshon and S. Sandon we prove Sandon's Conjecture for lens spaces equipped with the standard contact form. For that purpose we follow ideas of Givental and construct a quasimorphism G -> (R,+), i.e. a homomorphism up to bounded error, for G the universal cover of the identity component of the contactomorphism group of lens spaces. This quasimorphism, called a non-linear Maslov index, helps to understand the contactomorphisms of lens spaces. Apart from Sandon's Conjecture we prove: - that G is orderable, - that G can be equipped with unbounded bi-invariant metrics (which is important because any bounded bi-invariant metric must be trivial), - existence of non-displaceable pre-Lagrangian submanifolds of lens spaces, - that any contact form defining the standard contact structure on lens spaces has closed Reeb orbits (Weinstein Conjecture). In this talk I will present Sandon's Conjecture and outline its proof via non-linear Maslov index.
Mittwoch, den 11. Juli 2018 um 9:30-10:00 Uhr, in Mathematikon, INF 205, Konferenzraum, 5. Stock Mittwoch, den 11. Juli 2018 at 9:30-10:00, in Mathematikon, INF 205, Konferenzraum, 5. Stock
Der Vortrag folgt der Einladung von The lecture takes place at invitation by Prof. P. Albers