https://web3-math.univ-lyon1.fr/wikis/symp-foliations/ 2026-05-31T21:45:14+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 2017-08-25T15:17:43+00:00 Anonymous (anonymous@undisclosed.example.com) https://web3-math.univ-lyon1.fr/wikis/symp-foliations/doku.php?id=atsuhide_mori&rev=1503667063&do=diff Here are my comments on leafwise symplectic foliations: As is proved by David Martinez Torres, any closed manifold with codimension one 2-calibrated foliation has a closed transverse 3-dimensional submanifold which inherits a taut foliation with the text/html 2017-07-19T16:15:36+00:00 Anonymous (anonymous@undisclosed.example.com) https://web3-math.univ-lyon1.fr/wikis/symp-foliations/doku.php?id=confoliations&rev=1500473736&do=diff 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? text/html 2017-08-29T17:07:31+00:00 Anonymous (anonymous@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 2017-08-29T17:11:24+00:00 Anonymous (anonymous@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 2017-08-29T17:13:17+00:00 Anonymous (anonymous@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 2017-08-29T17:08:28+00:00 Anonymous (anonymous@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 2017-08-25T15:09:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://web3-math.univ-lyon1.fr/wikis/symp-foliations/doku.php?id=ioan_marcut&rev=1503666556&do=diff The topic related to the workshop that I am recently interested in is the study of “Poisson transversals” in Poisson geometry. These submanifolds play in Poisson geometry the role of symplectic submanifolds in symplectic geometry. A Poisson transversal is a submanifold in Poisson manifold that intersects the symplectic leaves transversally and symplectically. So far I managed to extend some of the simple results from the symplectic setting (in collaboration with Pedro Frejlich). text/html 2017-08-25T15:46:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://web3-math.univ-lyon1.fr/wikis/symp-foliations/doku.php?id=melanie_bertelson&rev=1503668815&do=diff 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? text/html 2017-07-19T16:16:30+00:00 Anonymous (anonymous@undisclosed.example.com) https://web3-math.univ-lyon1.fr/wikis/symp-foliations/doku.php?id=obstructions_to_existence_of_symplectic_foliations&rev=1500473790&do=diff Obstructions to existence. 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? text/html 2017-07-21T14:48:23+00:00 Anonymous (anonymous@undisclosed.example.com) https://web3-math.univ-lyon1.fr/wikis/symp-foliations/doku.php?id=pseudoholomorphic_curve_theory&rev=1500641303&do=diff Pseudo holomorphic curves Ever since Gromov published his '85 paper, pseudo-holomorphic curves have been one of the main tools in symplectic topology to prove “rigidity” results. Discussions related to holomorphic curves could be inspired by the following theorem in dimension 3. text/html 2017-08-29T17:14:57+00:00 Anonymous (anonymous@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 2017-07-05T18:27:49+00:00 Anonymous (anonymous@undisclosed.example.com) https://web3-math.univ-lyon1.fr/wikis/symp-foliations/doku.php?id=strong_symplectic_foliation&rev=1499272069&do=diff Strong symplectic foliation A regular symplectic foliation whose leafwise symplectic form arises as the restriction of a global closed 2-form. text/html 2017-08-29T17:05:58+00:00 Anonymous (anonymous@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 Anonymous (anonymous@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 … text/html 2017-08-25T16:06:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://web3-math.univ-lyon1.fr/wikis/symp-foliations/doku.php?id=yoshi_mitsumatsu&rev=1503670017&do=diff Since the meeting on Engel structures at AIM in April, I have been extensively working on the geometric construction of Engel structures (not h-principle type but really geometric, so it is just furnishing our fundamental understanding of Engel structures)and I myself feel what I have understood so far must have deep relations with Poisson structures or transverse contact foliations and so on.