(V. Ginzburg) Given a (possibly very degenerate) function on a foliated manifold, what is a sufficient condition for a leaf to have a critical point?

(M. Bertelson, V. Ginzburg) What about a foliated version of the Arnold conjecture? D. Castelvecchi made some progress in his thesis (2000), relating 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 L^2 Betti numbers of the foliation. Castelvecchi proved the Morse inequalities in this setting.

(M. Bertelson, R. Klaasse) Is there a foliated version of Seiberg-Witten theory providing obstructions to the existence of a leafwise symplectic structure for foliations of rank 4?

Given a foliation by surfaces on a 3-dimensional manifold, is there a way of obtaining 3-dimensional Seiberg-Witten by combining the vortex equation along the leaves coupled with some transverse differential condition? For mapping tori this is sketched by Salamon (1999). Assuming this is true, can this be used to say something about the foliations the 3-manifold admits? Can a similar scheme be used in higher dimensions to cook up invariants as suggested in the previous question?