Contact spectral invariants – cospin

The Arnold conjecture on fixed points of Hamiltonian diffeomorphisms motivated the discovery and development of some of the most powerful techniques of symplectic topology. Among these, generating functions and Floer homology. The method of generating functions is technically quite simple, being just based on classical Morse theory, but it only applies to some basic symplectic manifolds. On the other hand, Floer homology is a sophisticated infinite-dimensional Morse theory that integrates holomorphic curves techniques introduced by Gromov. It is technically quite challenging, but it gives a general framework that applies to very large classes of symplectic manifolds. Both theories have been used to define spectral invariants for Hamiltonian diffeomorphisms, one of the main tools to study (in particular) embedding problems (for instance the non-squeezing theorem), the Hofer metric and Hamiltonian dynamics. The construction of spectral invariants coming from generating functions (due to Viterbo) has been extended to the contact case by Sandon in 2009 and used to reprove the contact non-squeezing theorem. This new proof led to the discovery of a bi-invariant metric on the contactomorphism group and of the notion of translated points, that in turn allowed to formulate a contact analogue of the Arnold conjecture. While we are planning to continue to develop the theory of generating functions and its applications to contact rigidity, a second goal of the project is to construct contact spectral invariants using holomorphic curves, in order to obtain a more general framework to study contact rigidity. Finally, a third important ingredient of the project is to continue the research on spectral invariants in symplectic topology and Hamiltonian dynamics. Besides its intrinsic interest, we hope that any progress in this direction might have a contact analogue that would interact interestingly with the thematics described above.

The proof of the contact non-squeezing theorem using generating functions showed the key role played in this phenomenon by translated points of contactomorphisms. While generating functions are not known to exist on general contact manifolds, the notion of translated points is always well-defined and can be used to formulate a contact analogue of the Arnold conjecture. Motivated by this conjecture, Sandon is developing a Floer homology theory for translated points, which is almost ready in a relatively simple case (namely the «hypertight« case) and hopefully will serve as a starting point to construct contact spectral invariants and study global rigidity of contactomorphisms for more general contact manifolds than those which can be dealt with generating functions. In the meantime Leclercq and Mazzucchelli (independently) have continued the study of spectral invariants and their applications to symplectic topology and Hamiltonian dynamics, and Ferrand has worked with Limouzineau on the question of determining, by a constructive approach, which Legendrian embeddings can be realized (or not) by techniques of the kind «generating functions standard at infinity«. Finally, in collaboration with Ghiggini, Golovko and Rizell, Chantraine has developed a Floer homology theory for Lagrangian cobordisms, which relates the singular homology of a Lagrangian cobordism to a contact invariant (the Legendrian contact homology) of the endpoints. In a collaboration with Colin he uses this theory to find obstructions to the existence of certain positive loops of Legendrian submanifolds in hypertight contact manifolds proving, as a corollary, that such contact manifolds are orderable.

We hope that the Floer homology theory for translated points that is being developed will give us a tool to extend to the contact case (some aspects of) the theory of spectral invariants, in a more general framework than the one reacheable with generating functions. We also expect interesting interactions, that we are planning to explore, of this theory with other theories of Floer type in contact topology (contact homology, Legendrian contact homology, SFT, ...) and with the Floer homology theory for Lagrangian cobordisms that is studied by Chantraine and his collaborators. Still regarding relations between Legendrian submanifolds and contact transformations, Leclercq has started (with Seyfaddini) the study of rigidity of Legendrians under the action of contact homeomorphisms. In this project a new capacity appears, and we are planning to study its relations with spectral invariants. Chantraine and Colin's results should apply to a class of contact manifolds larger than hypertight contact ones as contact homology is now defined for general contact manifolds thanks to recent work of Pardon. While the projects mentioned so far mostly involve holomorphic curves techniques, we are planning to keep developing also the theory of generating functions. Indeed, we hope that the relative simplicity and different perspective of this technique (when applicable) might allow (as it has already been the case in the past) to detect new phenomena, that would be harder to notice among the technicalities of the other more sophisticated techniques.

The projects mentioned above led to:
- 4 publications in international refereed journals,
- 1 publication in a French refereed journal,
- 5 preprints available on the arXiv website.
They were also discussed during:
- 12 international mathematical conferences,
- 2 French mathematical conferences,
- numerous talks given in the maths department of many French and international universities.

Our proposal is in symplectic and contact topology, and has two main goals.

The first goal is to try 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.

Our 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 would use generating functions and/or holomorphic curves, trying to benefit also from the interactions between these two methods.

As a further important part of our proposal we also plan to 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.