Université de Lyon 1 (France)

4-8 September 2017

Organizers:

- Klaus Niederkrüger, Université de Lyon I
- Álvaro del Pino Gómez, ICMAT
- Francisco Presas, CSIC and ICMAT

The workshop will be devoted to the understanding of the global topological properties of symplectic foliations in higher dimensional manifolds. In particular, we want to study the question if there exists a similar dichotomy for such foliations as the one between taut/non-taut foliations in dimension 3. We want to understand the extent to which standard symplectic techniques apply in this new setting, under- stand how these foliations relate to contact structures, and explore new ways to construct examples.

Other perspectives and related open problems are also welcome.

The dichotomy between taut/non-taut foliations in dimension 3 extends (conjecturally) to higher dimensions by saying that a symplectic foliation is strong if the leafwise symplectic form arises from a global closed 2-form, and saying that it is weak otherwise. Certain symplectic techniques only extend (naively) to the first setting: Donaldson techniques and cohomological energy estimates for pseudoholomorphic curves do require closeness.

The following is a tentative list of potentially interesting topics for the workshop:

- Existence of (strong) symplectic foliations. One of the main problems of the theory is the lack of examples. The aim would be to try to find new constructions and study whether an h-principle can possibly hold in the weak symplectic case.
- Obstructions to existence. In the opposite direction, there are no known obstructions to the existence of strong symplectic foliations (apart from the obvious formal ones). Does a Novikov type theorem hold in this setting? What are some reasonable hypothesis for such a statement?
- The confoliation programme. The formal data underlying a symplectic foliation and a contact structure is the same. Is it possible to reproduce Eliashberg and Thurston’s confoliation result in higher dimensions? Is it perhaps simpler to carry it out if one assumes strongness?
- Pseudoholomorphic curve theory. In the strong case, under reasonable assumptions, moduli spaces of pseudoholomorphic curves should be compact manifolds endowed with singular foliations where the leaves correspond to the leafwise moduli spaces. What structure results for strong symplectic foliations can be obtained this way? What about classification results for fillings of contact foliations?
- Foliated Hamiltonian dynamics. What is the analogue of the Arnold conjecture in the strong symplectic case? Is there a meaningful foliated Lagrangian Floer theory? What about SFT?
- Transverse Hamiltonian dynamics. Strong symplectic foliations are not stable Hamiltonian usually, but there is a well defined notion of Reeb vector field. What can be said about its dynamics?