Timeline for What is so geometric about symplectic geometry?
Current License: CC BY-SA 4.0
35 events
| when toggle format | what | by | license | comment | |
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| Mar 24 at 3:17 | answer | added | Mozibur Ullah | timeline score: 3 | |
| Jun 2, 2021 at 12:43 | comment | added | alvarezpaiva | I think that what I object to is the "tight definition" of symplectic geometry. The study of the linear symplectic group, linear symplectic spaces, the Grassmannian of Lagrangian planes, the Maslov index and its variants, Hamiltonian systems, moment maps, etc., all that is symplectic geometry. | |
| Jun 2, 2021 at 12:29 | comment | added | alvarezpaiva | A huge portion of Riemannian geometry is just symplectic (the study of the Hamiltonian system described by its geodesic flow). Even the Levi-Civita connection is as easy to describe symplectically than it is to describe through the usual trick. See, for instance mathoverflow.net/questions/127319/… or mathoverflow.net/questions/256435/… | |
| Jun 2, 2021 at 10:39 | comment | added | Dan Fox | @alvarezpaiva: one confounding issue, relevant to the symplectic world, is where does volume fit? In the two-dimensional world area is part of geometry, whatever one means by geometry, In higher dimensions this is less clear. Perhaps one should speak also of "volumetric geometry". I think (maybe wrongly) of the "symplectic topology" terminology as due to Arnold. Note that Arnold/Khesin called their book "topological methods in hydrodynamics" not "geometric methods in hydrodynamics" and only write there of "geometry" when working with a metric on the diffeomorphism group. | |
| Jun 2, 2021 at 10:29 | comment | added | Dan Fox | @alvarezpaiva: I would put it this way - the debate about how to use "geometry" may or not be a worthwhile enterprise - I'm inclined to be cautious about assigning the word any tight definition - but then the use of "geometry" in "symplectic geometry" and "riemannian geometry" is very different in some senses. That's fine, the use of "geometry" in "algebraic geometry" is probably closer to the symplectic sense than the riemannian sense. | |
| Jun 2, 2021 at 9:54 | comment | added | alvarezpaiva | @DanFox, it is nowhere fixed that geometry must be local. For most of its history it wasn't and even today local results in differential geometry are not usually the stuff of top tier journals. The fact that there are no local invariants in symplectic geometry does not mean it is not geometric in the same way that the non-locality (or infinitesimal character) of theorems in projective geometry does not mean it is not geometric. | |
| Jun 2, 2021 at 9:40 | comment | added | Dan Fox | @alvarezpaiva: in projective differential geometry there is an underlying connection on a principal bundle with finite-dimensional structural group. This is not so for symplectic manifolds without further structure. In this sense symplectic manifolds are akin to differentiable manifolds and the study of their global properties is akin to differential topology, so could be called "symplectic topology". Whether it is worthwhile to make clean terminological distinctions between geometry and topology is another matter (where would combinatorics fit?). | |
| Jun 2, 2021 at 9:34 | comment | added | Dan Fox | This earlier question is closely related: mathoverflow.net/questions/283868/… | |
| Jun 1, 2021 at 21:26 | comment | added | Hollis Williams | What is geometric about von Neumann's continuous geometry, or the geometry of noncommutative spaces? | |
| Jun 1, 2021 at 18:58 | comment | added | Dev Sinha | A key example is the cotangent bundle, which is of course basic to study in differential geometry. I do that example along with Kahler manifolds give substantial geometric "bonafides". While this is at the opposite end of what the poster was asking about - axioms - I for one am satisfied by flowing from examples rather than axioms. | |
| Jun 1, 2021 at 17:09 | comment | added | Terry Tao | Personally, I view the subject of geometry as lying on a spectrum of rigidity, with topology at the loosest end of the spectrum and homogeneous spaces (such as Euclidean space) at the most rigid end. Riemannian geometry is roughly at the midpoint, Kahler and algebraic geometry is nearer to the rigid end, and symplectic geometry is nearer to the looser end. But they all share similar themes, for instance topics such as symmetries, invariants, localisation, embedding, or dependence on dimension are common to all. | |
| Jun 1, 2021 at 16:56 | comment | added | Terry Tao | @JannikPitt The geometric aspects of the study of fields (and other commutative algebras, rings, schemes, etc.) is more commonly known as "algebraic geometry". | |
| Jun 1, 2021 at 16:04 | comment | added | Ryan Budney | @joe: He goes into these sorts of ideas in a few places. But perhaps the place where he situates our modes of thinking about mathematics the most concretely is his "On proof and progress in mathematics" article. arxiv.org/abs/math/9404236 The "definition" I provide is an extrapolation from this -- I am not suggesting Thurston made this definition explicitly. But he certainly could have. | |
| Jun 1, 2021 at 9:11 | comment | added | Jojo | @Ryan Budney this is fascinating do you have a reference that makes this definiton? I could imagine interdisciplinary work on this in psychology or neuroscience, that would be really interesting | |
| Jun 1, 2021 at 7:54 | answer | added | alvarezpaiva | timeline score: 13 | |
| Jun 1, 2021 at 4:48 | review | Close votes | |||
| Jun 4, 2021 at 15:57 | |||||
| Jun 1, 2021 at 2:42 | history | became hot network question | |||
| Jun 1, 2021 at 2:15 | comment | added | Ryan Budney | A Thurston-inspired definition of "geometry" would come down to a measurement of what part of the brain you need to think about the subject. A subject is geometry in this sense if and only if you have to intensively use your visual cortex to think about the subject. I imagine there would be some variability from person to person and particular issues one might be engaged in thinking about, but I suppose this could be used as a starting point to make the word "geometry" more rigid. | |
| Jun 1, 2021 at 0:54 | answer | added | YHBKJ | timeline score: 20 | |
| May 31, 2021 at 22:22 | answer | added | B K | timeline score: 8 | |
| May 31, 2021 at 21:41 | comment | added | Jonny Evans | Another example: if you have a Lagrangian torus fibration, the action coordinates give you an integral affine structure on the base, which is extremely rigid and geometric. Skew symmetric forms are not what we're used to and you have to work a bit to understand the geometry they give you. | |
| May 31, 2021 at 21:39 | comment | added | Jonny Evans | Kahler manifolds have symplectic forms, and the symplectic geometry captures a lot of deformation-invariant geometry, like the Gromov-Witten invariants. For example, the existence of 27 lines on a smooth cubic surface is pretty uncontroversially a geometry theorem, and you still have it in symplectic geometry. | |
| May 31, 2021 at 21:38 | comment | added | Jonny Evans | Hamilton's equations and classical mechanics can seem very analytical. Symplectic geometry gives a very geometric way of thinking about them. For example, instead of "Hamilton's principal function" satisfying the Hamilton-Jacobi equation, you have a Lagrangian submanifold flowing under the Hamiltonian flow. Instead of a trajectory of the Hamiltonian system with given initial and terminal conditions, you have an intersection point between two Lagrangian submanifolds. | |
| May 31, 2021 at 21:31 | comment | added | RBega2 | I do feel that true geometry (or at least true differential geometry) should have some notion of curvature (or at least flatness should be a non-trivial condition). In this sense, symplectic geometry does seem closer in spirit to topology. | |
| May 31, 2021 at 21:15 | comment | added | Sylvain JULIEN | One could also wonder what a number is. Are quaternions genuine numbers? Their usefulness in physics gives them a "real" legitimity, even though Pascal said "les nombres imitent l'espace, qui est de nature si différente"... | |
| May 31, 2021 at 20:13 | comment | added | Ryan Budney | I think it's fair to say mathematicians use the word "geometry" generously. Traditionally there hasn't been a hard-and-fast rule for saying when something isn't geometric. You could extend your line of questioning further. Much of Riemannian geometry aligns with the study of PDEs. Does that mean PDEs should be viewed as the central to geometry? Or should we say much of Riemannian Geometry isn't geometric? Ultimately there aren't any strict rules applied to subject-area titles. I think we have a thread here somewhere, complaining about "K-theory" as a title, probably many others. | |
| May 31, 2021 at 19:58 | comment | added | Jannik Pitt | @SylvainJULIEN What is a "space"? Is then e.g. field theory also geometry? | |
| May 31, 2021 at 19:42 | comment | added | Sylvain JULIEN | To me geometry is the study of the properties of some kind of space, any set endowed with a sufficiently interesting structure deserving to be called that way. | |
| May 31, 2021 at 19:39 | comment | added | alvarezpaiva | Is projective geometry geometric enough? There are no angles, no distances, no lengths not even areas and volumes like in symplectic geometry. Perhaps you need to look at applications of symplectic and contact geometry to Riemannian geometry, optics, convex geometry, integral geometry, etc. so that you see the geometry come out. Just the basic definitions and the elegance of the Hamiltonian formalism won't give you any insight. | |
| May 31, 2021 at 19:13 | history | edited | LSpice | CC BY-SA 4.0 |
Proofreading
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| May 31, 2021 at 18:53 | comment | added | Ryan | A symplectic form is a closed non-degenerate 2-form. You can interpret this as a way of measuring 2-dimensional area for infinitesimal surface pieces. One is led to symplectomorphisms, as you say, which turn out to be more restrictive then asking for "volume preserving" maps (e.g. non-squeezing theorem). So there is some geometric content. | |
| May 31, 2021 at 18:50 | comment | added | RBega2 | It's not my area of research, but from the outside something like Gromov's non-squeezing theorem seems like a very geometric result. | |
| May 31, 2021 at 18:49 | history | edited | Jannik Pitt | CC BY-SA 4.0 |
added remark concerning symmetries
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| May 31, 2021 at 18:48 | comment | added | Abdelmalek Abdesselam | Klein's Erlangen Program essentially equates geometry with invariant theory of some subgroup (the principal group) of the general linear group (the comprehensive group). Here the principal group would be the symplectic group. I think the question is related to extending Klein's dictionary from invariant theory to differential invariant theory. | |
| May 31, 2021 at 18:40 | history | asked | Jannik Pitt | CC BY-SA 4.0 |