Talk:Symplectic topology

Discussion

This article does appear to have resulted in some clearing of the brushwood out of the forest - but I still find the need to ask the 'obvious question'...

I am a physicist (who graduated back in the days when they still took examinations on 3-legged stools :-). I've done a course on classical mechanics, and know the relevance of phase space and Liouville's theorem to statistical mechanics.

What new insights, mathematical or (preferably) physical, will symplectic topology in general and this article in particular offer me? (Or is this still a conversation piece between Mathematicians that others are not really bright enough to share?) Linuxlad 08:25, 9 Apr 2005 (UTC)

PS Nonetheless, the article is welcome

I'm a novice Wikipedia editor who just got his Ph.D. in symplectic topology. I am interested in developing this area of Wikipedia a bit further. The pseudoholomorphic curve article is a start; next is Gromov-Witten invariants. There is still much work to do in connecting the math pages with the string theory pages. This reflects the larger gap in language between the math and physics communities. IMHO Wikipedia could be a great help in closing this gap.--Joshuardavis 18:02, 23 July 2005 (UTC)[reply]

Symplectic topology is not the same as symplectic geometry

I don't think that symplectic topology and symplectic geometry should be used as synonyms.

I have never heard anyone say that work with moment maps etc. is symplectic topology. It is, however, quite naturally an example of symplectic geometry.

I would draw the line as follows : symplectic topology is interested in the various problems of a differential topology nature that one can ask about a symplectic manifold. For instance, I would group under this heading all questions of embeddings (of Lagrangians, of symplectic submanifolds, etc.)

Symplectic geometry is, in my opinion, the larger field that includes symplectic topology as well as all symplectic problems that have nothing to do with topology (e.g. how many torus group actions can you have).

In this sense, symplectic geometry has a fairly long history, whereas symplectic topology really came into its own only with Gromov's pseudoholomorphic curve paper.

Any thoughts on this?

SammyBoy 23:22, 30 November 2005 (UTC)[reply]

Your remarks are quite consistent with my experience. In particular, the examples of pseudoholomorphic curves as symplectic topology and moment maps as symplectic geometry are accurate. On the other hand, it is not clear to me that symplectic topology is a subset of symplectic geometry; the former is the study of soft properties of symplectic manifolds, and the latter is the study of rigid properties? Also, given that subjects within math are ill-defined and always reconfiguring, I'm not sure it's useful to worry about the distinction. Joshuardavis 01:52, 2 December 2005 (UTC)[reply]

I agree that the two terms have slightly different meanings. However, I also think there is a considerable overlap between the two subjects, perhaps more so than in the case of differential geometry vs. differential topology. Perhaps it would be a good idea to rename this page symplectic geometry and topology and discuss the differences between the two. -- Fropuff 16:30, 2 December 2005 (UTC)[reply]

By the way, I have changed the redirect on Symplectic geometry to point to Symplectic topology rather than Symplectic manifold. Joshuardavis 17:15, 14 January 2006 (UTC)[reply]

Well, it used to point here. I didn't catch that someone changed it to the manifold page. -- Fropuff 01:13, 15 January 2006 (UTC)[reply]

Until such time as it is deemed necessary to distinguish symplectic topology from symplectic geometry more carefully, I added a short parenthetical note at the beginning indicating that the two are not synonymous. (I also clarified Darboux's Theorem. It's not that it was stated incorrectly as such, but it was a little unclear since a locally isomorphic structure doesn't really involve the notion of a "pair of manifolds".) VectorPosse 09:23, 13 July 2006 (UTC)[reply]

Thanks for the excellent thoughts. I just noticed that there is a Moment map page. We should see about writing a little paragraph about "classical" symplectic geometry (Moment maps, symplectic reductions, Delzant polytopes, etc.) Unfortunately, I do symplectic topology and have only a sketchy knowledge of these things. Anyone with some expertise? A big picture of the field may help give a better sense of what is important and what isn't. -- SammyBoy 08:12, 16 September 2006 (UTC)[reply]