Timeline for Is differential topology a dying field? [closed]
Current License: CC BY-SA 4.0
36 events
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| Jun 5, 2019 at 3:05 | review | Reopen votes | |||
| Jun 5, 2019 at 9:08 | |||||
| Jun 1, 2019 at 20:26 | comment | added | user140765 | @NajibIdrissi so Najib Idrissi do you agree that you accused me of using words wrongly before I accused you of that or not? A definition that is natural for you is not natural for everyone. | |
| May 30, 2019 at 23:04 | history | rollback | Wojowu |
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| May 30, 2019 at 22:52 | history | edited | user44143 | CC BY-SA 4.0 |
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| S May 30, 2019 at 21:40 | review | Reopen votes | |||
| May 30, 2019 at 23:30 | |||||
| May 30, 2019 at 21:40 | history | closed |
Timothy Chow mme Deane Yang LSpice Todd Trimble |
Opinion-based | |
| May 30, 2019 at 19:48 | history | became hot network question | |||
| May 30, 2019 at 19:20 | comment | added | LSpice | @TimothyChow, are questions about dying fields dying? | |
| May 30, 2019 at 19:20 | review | Close votes | |||
| S May 30, 2019 at 21:40 | |||||
| May 30, 2019 at 19:03 | comment | added | Timothy Chow | Questions about whether a field is dying or dead seem too opinion-based for MO. Compare mathoverflow.net/questions/332281/is-the-field-of-q-series-dead Voting to close. | |
| May 30, 2019 at 18:52 | answer | added | Ian Agol | timeline score: 43 | |
| May 30, 2019 at 18:08 | comment | added | user140765 | I am also inferring that you accused me of using the words wrongly before I accused you of that (essentially you applied your definition of differential topology to my statement, which indeed does not make much sense given your definition). | |
| May 30, 2019 at 18:06 | comment | added | user140765 | @NajibIdrissi OK, I apologize for my wrong inference then (which resulted from me applying my definition of differential topology to your statement). You apparently have a different definition of differential topology, which I think is also reasonable, and your statement is compatible with that definition. | |
| May 30, 2019 at 18:03 | comment | added | Najib Idrissi | @kartop_man Then you inferred wrong. What is natural to you is not natural to everyone. Riemannian geometry is closer to differential geometry (and yes, the name is usually symplectic geometry, and yes, if you ask people who work in the field, many will tell you that they prefer the name symplectic topology). | |
| May 30, 2019 at 18:02 | comment | added | user140765 | I think that differential topology is about studying the discrete invariants of smooth structures on topological manifolds. Symplectic topology is not necessarily a part of differential topology according to this definition (although there are some connections, something like the work of Abouzaid exploring how smooth structures on a manifold are reflected in symplectic structures on the cotangent bundle). This definition is probably not the only reasonable one, but I do believe that it is reasonable too. | |
| May 30, 2019 at 17:56 | comment | added | user140765 | Also, could you please explain to me whether Riemannian geometry is also to be considered as differential topology or not? Riemannian manifolds are smooth manifolds, and Riemannian metrics are tensors. | |
| May 30, 2019 at 17:54 | comment | added | user140765 | @NajibIdrissi maybe I am misunderstanding you. What you did say is that "a symplectic form is a differential form". I inferred from this that you think that symplectic topology is about studying these symplectic forms up to the natural equivalence relation on random differential forms (which is smooth isotopy). What I said is that most of the interesting phenomena in symplectic topology are not about this (though some are). | |
| May 30, 2019 at 17:43 | comment | added | Najib Idrissi | @kartop_man I do not appreciate you putting words in my mouth. At what point did I claim that symplectic geometry was about smooth isotopies? What do you think differential topology is? | |
| May 30, 2019 at 16:11 | vote | accept | James Baxter | ||
| May 30, 2019 at 15:14 | answer | added | Mark Grant | timeline score: 34 | |
| May 30, 2019 at 14:36 | comment | added | Nik Weaver | +1 "we can just call each other wrong and stop there" | |
| May 30, 2019 at 13:59 | comment | added | user140765 | @WarlockofFiretopMountain OK, the criticism regarding the deadness of Donaldson's theory I accept. The first sentence I do not completely agree with, one should define what is "a topological result" first. There are for example symplectomorphisms which are smoothly isotopic to the identity but are not symplectically isotopic to the identity. Is this "a topological result"? Probably. I do not think that it is a differential topological result though. Either way, I believe that this is not a mathematical discussion, but a linguistic one, so we can just call each other wrong and stop there. | |
| May 30, 2019 at 13:54 | comment | added | Overflowian | @kartop_man If you prove a topological result in the category of Riemmanian Manifold that is righteously a Differential Topology result. Symplectic topology and Contact topology, are specific (big) fields of Differential Topology that earned their own name. Btw Donaldson theory is not as dead as you depict it, there is still people working on instantons on manifolds with special holonomy or in higher dimension or on extensions and applications of Instanton Floer Homologies and I believe there is more. | |
| May 30, 2019 at 13:43 | comment | added | Neal | @MarkGrant That looks like an answer to me. :) | |
| May 30, 2019 at 13:15 | comment | added | user140765 | @NajibIdrissi well, in symplectic topology most of the activity is not studying these differential forms up to smooth isotopies (but rather up to symplectomorphisms, or Hamiltonian isotopies). By your logic, Riemannian geometry is also differential topology because Riemannian manifolds are smooth and a metric is a rank 2 tensor. Or am I misunderstanding something? | |
| May 30, 2019 at 13:05 | comment | added | Najib Idrissi | @kartop_man I don't really understand your last comment. A symplectic manifold is smooth, and a symplectic form is a differential form... Yet you use the words as if they were opposite. | |
| May 30, 2019 at 12:56 | comment | added | user140765 | @FrancescoPolizzi also I do not believe that symplectic topology is a part of differential topology. A question of definitions, you could say, but to me "differential" means continuously differentiable, smooth or real-analytic, definitely not symplectic. | |
| May 30, 2019 at 12:55 | comment | added | user140765 | @FrancescoPolizzi from my limited perspective, Donaldson theory is indeed not a dying field but a dead field. SW have put it out of business, I think. Or are there some interesting topological questions in which it is easier to use the Donaldson invariants? | |
| May 30, 2019 at 12:18 | comment | added | James Baxter | Ah, that does make a lot of sense @MarkGrant | |
| May 30, 2019 at 12:17 | comment | added | Mark Grant | Probably it is the blanket term "differential topology" which is dying, as people use more specific terms to describe different aspects of the study of smooth manifolds and maps. | |
| May 30, 2019 at 12:04 | comment | added | Todd Trimble | I don't know the answer to this question. Perhaps it is self-centered to say, but it's long been a pipe dream of mine to develop further a rapprochement between differential topology and the geometry of higher categories, as has been partially explored in papers like 2-Tangles by Baez and Langford. | |
| May 30, 2019 at 12:04 | comment | added | Francesco Polizzi | "Differential topology" in a broad sense includes knot theory, Seiberg-Witten theory, Donaldson theory, symplectic topology. How can it be a dying field? | |
| May 30, 2019 at 11:58 | comment | added | Denis Nardin | I guess it depends on what do you mean with differential topology. I have the impression that several subareas are very active (e.g. surgery theory) but I'm not very familiar with the field as a whole | |
| May 30, 2019 at 11:53 | history | edited | James Baxter | CC BY-SA 4.0 |
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| May 30, 2019 at 11:48 | history | edited | James Baxter | CC BY-SA 4.0 |
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| May 30, 2019 at 11:42 | history | asked | James Baxter | CC BY-SA 4.0 |