A codimension one foliation F of a 3-manifold M is said to be taut if for every leaf λ of F there is a circle transverse to F which intersects λ.

This definition, which is of topological nature, can be alternatively characterised as follows:

Theorem (Rummler, Sullivan). For a smooth transversely orientable codimension one foliation F in a closed, orientable, smooth manifold M, the following conditions are equivalent:

1. F is taut,
2. there is a flow transverse to F which preserves some volume form on M,
3. there is a closed 2-form on M that is an area form over each leaf,
4. there is a Riemannian metric on M for which the leaves of F are least area surfaces.

Taut foliations in dimension 3 present a rich geometrical and topological behaviour. The following theorem follows from the theory of minimal surfaces:

Theorem (Novikov, Rosenberg). Let F be a taut foliation on a closed 3-manifold M. The following statements hold:

1. any loop transverse to F represents a non trivial class in homotopy,
2. any leaf of F pi_1 injects into the ambient space,
3. M is irreducible.

Gabai was able to relate the construction of taut foliations to decompositions of the ambient manifold into sutured pieces and to the construction of essential laminations. In particular:

Theorem (Gabai). Let M be a closed irreducible 3-manifold with non zero second real cohomology. Then M admits a taut foliation.

The existence of taut foliations in rational homology spheres is a topic of active research. An outstanding conjecture in 3-dimensional topology states that a rational homology sphere admits a taut foliation if and only if it is not an L-space and if and only if its fundamental group is left orderable.

Taut foliations also relate to contact structures. The Eliashberg-Thurston confoliation construction shows that any taut foliation (or more generally, a foliation without a Reeb component) can be approximated by a tight (and in particular, fillable) contact structure.

This rich theory contrasts with the following facts:

Theorem (Thurston). Let M be a closed manifold with zero euler class. Then M admits a foliation.

Theorem (Meigniez). Let M be a closed manifold with zero euler class of dimension at least 4. Then M admits a foliation by dense leaves (which is, in particular, taut).

The purpose of the workshop is to explore whether one may be able to define interesting classes of foliations in higher dimensions by taking a symplectic viewpoint. We say that a foliation is symplectic if it admits a leafwise symplectic form. If this form arises from a global closed 2-form we say that the foliation is strong or 2-calibrated. Certain symplectic techniques only extend (naively) to the strong 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?

The following is a list of the particular topics and questions brought up by the participants.

• Differential operators and foliations

• Transverse submanifolds and foliations

• Existence of symplectic foliations

• The confoliation programme

• Deformations of symplectic foliations

• Foliated symplectic topology