Milena Pabiniak (University of Cologne) will speak in the geometry seminar on the 24th of April. The talk will take place in the Salle de Profs (9th floor, NO) at 11am. Milena’s title is
Understanding the contactomorphism group of lens spaces via a non-linear Maslov index and her abstract is below.
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.
In this talk I will describe joint work with G. Granja, Y. Karshon and S. Sandon in which we construct a quasimorphism G -> (R,+), i.e. a homormorphism up to bounded error, for G the universal cover of the identity component of the contactomorphism group of lens spaces.
This quasimorphism helps to understand the contactomorphisms of lens spaces. In particular we prove:
– Sandon’s Conjecture for lens spaces,
– 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.
This construction is a generalization of work of Givental on real projective spaces.
If time permits, I will discuss the possibility of generalizing these ideas even further: to prequantizations of symplectic toric manifolds.