symp-foliations

A) Is there a particular technique/theory/topic/result that, in your opinion, should be explained during the workshop? If there is, please let us know what its role is within the theory of symplectic foliations. Are there relevant open questions that this technique might be able to answer? Would you be willing to give a talk/minicourse about it?

- Some aspects of the theory of h-principles are likely to be relevant to understand the question of existence and classification of symplectic foliations. I can give an introduction to the subject but there are lots of other people that would do that very well (including you of course).

- As mentioned below, Seiberg-Witten invariants might be relevant to non-existence but that is very hypothetical.

- Some aspects of Connes' non-commutative integration theory might play a role when trying to build foliated invariant (cf. Castelvecchi's foliated Morse inequalities mentioned below) but that might not be of general interest.

B) What questions do you find interesting about symplectic foliations? (They do not need to be of general interest. Very specific questions are also welcome, since they might help to guide the discussion sessions.)

I find many questions very interesting but here are a few questions that interest me more closely :

- I am interested in the question of existence of symplectic foliations. In particular is it possible to prove an h-principle for codimension one symplectic foliations on closed manifolds (it has been proven for open foliations by Fernandes and Frejlich), i.e. given a codimension one symplectic hyperplane field, can it be deformed amongst such hyperplane fields to a codimension one symplectic foliation? Marius Crainic believes that the existence problems for codimension one symplectic foliations and for contact structures (both share the same formal data) are equivalent. On the other hand, there are obstructions to existence of symplectic structures on closed 4-dimensional manifolds coming from Seiberg-Witten theory and I wonder if there might be a remnant of that for foliations of leaf-dimension 4 on closed 5-manifolds.

- I believe it would be very interesting to develop the symplectic topology of symplectic foliations. In particular, address the question “is there some interesting new phenomena appearing that intertwines dynamical aspects of foliations with symplectic rigidity?” (this is quite vague). D. Castelvecchi made some progress in his thesis (2000) towards establishing a foliated version of the Arnold conjecture that mixes symplectic rigidity and Connes' noncommutative integration theory. The optimal statement is that given a symplectic foliation on a closed manifold and a transverse measure, the measure of the set of fixed points of a (generic) leafwise Hamiltonian flow is bounded below by the sum of the foliated L² Betti numbers of the foliation. Castelvecchi proved in fact the Morse inequalities in this setting.

- Duality between tangentially symplectic foliations and transversally symplectic foliations or constant rank closed 2-forms. A foliation transverse to a transversally symplectic foliation of of course symplectic but how about the converse? It is for instance unclear to me whether an h-principle holds for transversally symplectic foliations on open manifolds as is true for symplectic foliations. In the case where the leaf space is four-dimensional, it might be that there are remnants of the obstruction for existence of symplectic structures coming from Seiberg-Witten. I am not aware of anyone having tried to “foliate” Seiberg-Witten. There are some progress towards Seiberg-Witten for transversally symplectic foliations of dimension 4 due to Kordyukov, Lejmi and Weber.