https://web3-math.univ-lyon1.fr/wikis/symp-foliations/ 2026-04-05T22:36:47+00:00 https://web3-math.univ-lyon1.fr/wikis/symp-foliations/ https://web3-math.univ-lyon1.fr/wikis/symp-foliations/lib/tpl/dokuwiki/images/favicon.ico text/html 2017-08-29T17:14:57+00:00 alvaro (alvaro@undisclosed.example.com) https://web3-math.univ-lyon1.fr/wikis/symp-foliations/doku.php?id=start&rev=1504019697&do=diff Foliations in dimension three 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: text/html 2017-08-29T17:13:17+00:00 alvaro (alvaro@undisclosed.example.com) https://web3-math.univ-lyon1.fr/wikis/symp-foliations/doku.php?id=existence_of_symplectic_foliations&rev=1504019597&do=diff Existence of symplectic foliations * (R.L. Fernandes, F. Presas, G. Meigniez) Conjecture: Let M be a closed, odd-dimensional, simply--connected manifold. Then it does not admit a codimension-1 strong symplectic foliation. It is known by work of Mitsumatsu that the 5-sphere does admit a weak symplectic foliation. In dimension 3, the conjecture is true due to Novikov's theorem. text/html 2017-08-29T17:11:24+00:00 alvaro (alvaro@undisclosed.example.com) https://web3-math.univ-lyon1.fr/wikis/symp-foliations/doku.php?id=differential_operators_and_foliations&rev=1504019484&do=diff Differential operators and foliations * (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… text/html 2017-08-29T17:08:28+00:00 alvaro (alvaro@undisclosed.example.com) https://web3-math.univ-lyon1.fr/wikis/symp-foliations/doku.php?id=foliated_symplectic_topology&rev=1504019308&do=diff Foliated symplectic topology * What is the precise foliated analogue of Gromov's compactness for symplectic foliations?. Energy bounds seem to require strongness. Can pseudoholomorphic curve techniques still be carried out in the weak setting? Pancholi and Venugopalan have proven an analogue of McDuff's classification result of ruled symplectic 4-manifolds. text/html 2017-08-29T17:07:31+00:00 alvaro (alvaro@undisclosed.example.com) https://web3-math.univ-lyon1.fr/wikis/symp-foliations/doku.php?id=deformations_of_symplectic_foliations&rev=1504019251&do=diff Deformations of symplectic foliations * (E. Miranda) It would be interesting to understand better how to characterise those symplectic leaves that survive under perturbations of the Poisson structure/symplectic foliation. Are there KAM--type theorems? What do they say about the foliation? text/html 2017-08-29T17:05:58+00:00 alvaro (alvaro@undisclosed.example.com) https://web3-math.univ-lyon1.fr/wikis/symp-foliations/doku.php?id=the_confoliation_programme&rev=1504019158&do=diff The confoliation programme * In dimension 3 it is known, due to Eliashberg-Thurston, that any foliation can be C0-approximated by a positive and a negative contact structure. Is the same true in higher dimensions for a symplectic foliation? * In dimension 3 tautness of the foliation implies tightness of the contact structure. In higher dimensions, being strong symplectic implies that any approximation (if it exists) is fillable and thus tight as well. text/html 2017-08-29T17:01:15+00:00 alvaro (alvaro@undisclosed.example.com) https://web3-math.univ-lyon1.fr/wikis/symp-foliations/doku.php?id=transverse_submanifolds_and_foliations&rev=1504018875&do=diff Transverse submanifolds and foliations * (Ioan Marcut) Strong symplectic foliations of codimension-1 admit strong symplectic divisors as a consequence of Donaldson theory. What about foliations of higher codimension? (It is worth pointing out that in codimension-1 one obtains a non-singular foliation because of the symplectic analogue of Bertini's theorem: singular divisors generically appear in real families of dimension at least 2.) What about weak symplectic foliations, to which Donaldson …