Research
On s'habitue, c'est tout.
I'm mostly interested in higher dimensional contact geometry. In contrast to the 3-dimensional case, this is a relatively unexplored area with many basic questions unanswered. Recently though this direction has attracted a lot of attention by several new and unexpected discoveries (e.g. the notion of loose Legendrian knots and the existence of contact structures on any almost contact manifold).
From 2010 to 2013, my research has been partially funded by the ANR project Higher dimensional contact topology.
Video lectures: I have recorded a few video lectures on the proof of non-displaceability of exact Lagrangian submanifolds given by Gromov in his paper from 85. I would be interested in any feedback. You can watch them here:
Trivia: My research area is a subdomain of
Symplectic geometry. Apart from the mathematical
definition, I do not have any clue what "symplectic"
actually means, but on a trip to Asia, I learned that the Chinese
character for "symplectic" is
"辛", because that
character is pronounced sin-plectic.
The standard meaning of "辛" is "hot, spicy", so that at least in China I'm a hot geometer...
Nonfillability and Overtwistedness
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For many years my work was centered around non-fillability questions, and ideas on how to generalize the notion of overtwisted to higher dimensions. The bLob (being an acronym of "bordered Legendrian open book") was such a conceivable generalization of the overtwisted disk. First examples of closed manifolds containing a bLob were found by Francisco Presas, and soon afterwards it was discovered that any contact manifold can be modified to admit such a submanifold. By now due to the work of Borman-Eliashberg-Murphy a definition of overtwistedness is known that implies flexibility which settles the question of overtwistedness in higher dimensions at least on the theoretical level. An article by Casals-Murphy-Eliashberg then shows that many of the previous conjectural definitions of overtwistedness are in fact equivalent to the one by Borman-Eliashberg-Murphy. The general opinion these days is that the most workable definition of overtwistedness consists in regarding stabilizations of overtwisted 3-manifolds first introduced in the joint paper with Fran Presas. |
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Group actions on contact manifolds, symplectic orbifolds, open books
A second part of my work uses more geometric methods like group actions, branched covers, resolution of orbifolds, and open books.
Articles:
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All articles may also be found on the arXiv .
Other
People
- Frédéric Bourgeois
- Hansjörg Geiges
- Paolo Ghiggini
- Otto van Koert
- Sam Lisi
- Patrick Massot
- Emmy Murphy
- Ferit Öztürk
- Federica Pasquotto
- Olga Plamenevskaya
- Francisco Presas Mata
- Ana Rechtman
- Felix Schlenk
- András Stipsicz
- Chris Wendl
References
- Yakov Eliashberg Classification of overtwisted contact structures on 3-manifolds Invent. Math. 98 (1989), no. 3, 623-637.
- Francisco Presas A class of non-fillable contact structures Geom. Topol. 10 (2006), 1373-1389.
- Emmy Murphy Loose Legendrian Embeddings in High Dimensional Contact Manifolds arXiv (2012) 1201.2245
- Matthew Strom Borman Yakov Eliashberg Emmy Murphy Existence and classification of overtwisted contact structures in all dimensions Acta Math. 215 (2015), no. 2, 281-361
- Roger Casals Emmy Murphy Francisco Presas Mata Geometric criteria for overtwistedness arXiv math.SG:1503.06221