Talk:Symplectic geometry

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Too narrow[edit]

Come now. Saying that symplectic geometry is a subbranch of differential geometry is being myoptic at best. This topic is far more general than that and needs a rewrite from a more general perspective. Unfortunately, I am not the right person to do this. Bill Cherowitzo (talk) 17:43, 8 August 2014 (UTC)

The article says "branch of differential geometry and differential topology". To me that's a pretty good description, and not at all insulting to symplectic geometry. Would you rather say that it's a branch of mathematics? A branch of geometry/topology? A geometric framework for classical physics, which turns out to be topological and pertinent to contemporary physics? I'm not sure what you want. Mgnbar (talk) 19:44, 8 August 2014 (UTC)
I probably misunderstand the term, but it certainly sounds like the differential and topological aspects need not be inherent to the geometry. Elliptic geometry is valid even with only an algebraic definition, and this is presumably no different. So, "a branch of geometry" sounds right (to me as a layman, anyway). —Quondum 01:12, 9 August 2014 (UTC)

Symplectic geometry is also studied by algebraic geometers and the differential and topological aspects play no role in that study. You would not get a hint of this by reading this article. The classical example does have additional structure and this has been exploited and generalized by differential geometers/topologists. This is also what makes the theory useful in applications to physics. However, to say that differentiable manifolds are the natural setting for the subject strikes me as having the tail wag the dog. I am not advocating the removal of any material (with the possible exception of the last sentence which I think is a misinterpretation of Weyl's comment), rather I'm looking for a broader framework in which the current article would comfortably sit. Bill Cherowitzo (talk) 04:12, 9 August 2014 (UTC)

From my viewpoint, symplectic geometry/topology is the study of symplectic manifolds. A symplectic manifold is a smooth manifold equipped with a symplectic form. So it is a branch of topology. But people quickly add additional information such as almost-complex structures and metrics, so that it becomes geometry. But a lot of that geometry ends up being topologically determined anyway. The interactions with algebraic geometry come from Kahler examples. For example, the Gromov-Witten invariants are defined using a metric, but end up being symplectic invariants, and related to enumerative invariants in algebraic geometry. What am I missing? Perhaps you could clarify your viewpoint more, to someone with my viewpoint. Mgnbar (talk) 07:52, 9 August 2014 (UTC)
After re-reading Quondum's post, I'm wondering if there's a reference (preferably a PDF online, or a canonical textbook) that explains how to get symplectic geometry on a purely algebraic level, without an underlying smooth manifold. Learning is fun. Mgnbar (talk) 08:00, 9 August 2014 (UTC)
A smooth complex projective variety is a symplectic manifold, so one could use either the algebraic or symplectic techniques to characterize the object. A tutorial discussing the two points of view for toric varieties is at Symplectic Toric Manifolds. --Mark viking (talk) 08:30, 9 August 2014 (UTC)
So far I'm not convinced. We're talking about objects (complex projective varieties, toric varieties) that exist in the intersection of symplectic geometry/topology, complex algebraic geometry, and complex analysis (and maybe geometric analysis, string theory, ...). I alluded to this above, in mentioning that many of the examples in symplectic geometry are Kahler. But that doesn't mean that all of symplectic geometry is Kahler, or that symplectic geometry should be regarded as existing independently of symplectic manifolds.
Maybe I'm over-interpreting this thread. Are we simply saying that the article should mention more connections to other subjects, including algebraic geometry? Because that request is totally reasonable, and more manageable than rewriting the article. Mgnbar (talk) 15:37, 9 August 2014 (UTC)
I don't think a rewrite is needed; the information in already there is good. Some expansion would good to indicate connections with other approaches and applications, e.g., algebraic geometry, geometric combinatorics, optics, gauge field theory, etc. I agree that I have never seen symplectic geometry treated without mentioning a symplectic manifold. The symplectic 2-form has to be there in some guise. --Mark viking (talk) 17:06, 9 August 2014 (UTC)

After touring some of the related pages, I think that I'm going to have to settle for a hatnote here. What WP calls Symplectic vector space, I would call symplectic geometry, following Artin (Geometric Algebra) and several other authors with an algebraic leaning. Perhaps the term "polar space" has gained more popularity over the last 50 years for this algebraic formulation, so that one doesn't see the term symplectic geometry in this sense as often these days. Bill Cherowitzo (talk) 18:59, 9 August 2014 (UTC)

(Disclaimer: much of what I'm saying here is pure speculation due to my lack of familiarity with the subject matter.) Under "symplectic geometry", I would assume a geometry as characterized by the Erlangen program, in terms of a set with a group of transformations that might be described as symplectic, in the sense that a symplectic bilinear form is preserved by the group. This is a pure algebraic approach, but does allow a fairly precise definition of the term "geometry" in this restricted sense. This admits a rich array of geometries, and although it does not extend to general geometries on manifolds, it gives excellent prototypes. Geometries over finite fields are interesting, and since a symplectic bilinear form is a pure algebraic concept and is defined on vector spaces over any finite field, it would feel a bit odd for geometries defined in terms of symplectic bilinear forms on finite geometries to be excluded, just because the concepts of continuity and topology have little use. But of course, if the term symplectic geometry is spoken for by a particular mathematical discipline that is only interested in particular types of geometries, WP could reasonably choose to use it in this way, though to me, the natural title for this article would be Symplectic geometry of a manifold. —Quondum 20:32, 9 August 2014 (UTC)
The group that you're talking about is the symplectic group. As far as I've seen and can recall, treatments of symplectic geometry (such as McDuff and Salamon) do talk about symplectic vector spaces, but only in preparation for symplectic manifolds. (Their tangent spaces are symplectic vector spaces.) So I do think that the term "symplectic geometry", as it is actually used in the mathematics community, describes the material surveyed by this article. But more material and more connections to other subjects would be great. Mgnbar (talk) 21:28, 9 August 2014 (UTC)
I've just realized that I've missed a major point. This article is about an area of study, not about a mathematical object. It studies the objects called symplectic manifolds. So you can cancel everything I've said, and I apologize for any confusion that I may have added. Like I said "I probably misunderstand the term ...".
I know it is implied in the description, but your description ("Their tangent spaces are symplectic vector spaces") is a nice description of a symplectic manifold, and both this and the Symplectic manifold article might benefit by having it. —Quondum 23:33, 9 August 2014 (UTC)