ANR Project: COSPIN — Spectral invariants in contact geometry (ANR-13-JS01-0008-01)
Starting date: Jan. 1, 2014. Duration: 4 years.

This project has two main goals. The first goal is to generalize to contact topology, starting from the contact spectral invariants recently introduced by Sandon, as much as possible of what is known about spectral invariants in symplectic topology and Hamiltonian dynamics (in particular, what has been studied by Leclercq and Mazzucchelli) and use this to get applications to contact rigidity phenomena such as contact non-squeezing, orderability, translated points, bi-invariant metrics and quasimorphisms on the contactomorphism group.
The second main goal is to find new interplays between the above contact rigidity phenomena (that can be seen as global rigidity of contactomorphisms) and the global rigidity of Legendrian submanifolds as recently studied among others by Chantraine.

In order to carry out this program we will use generating functions and/or holomorphic curves, trying to benefit also from the interactions between these two methods.

As a further important part of this project we will also continue the study of symplectic spectral invariants and their applications to geometric and dynamical properties of Hamiltonian diffeomorphisms and to the geometry of Lagrangian submanifolds. Besides its intrinsic interest, we see this part also as common background and source of inspiration for the two main goals described above.