Timeline for If a two variable smooth function has two global minima, will it necessarily have a third critical point?
Current License: CC BY-SA 4.0
42 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Oct 20, 2021 at 5:31 | history | bounty ended | Jyrki Lahtonen | ||
| S Oct 20, 2021 at 5:31 | history | notice removed | Jyrki Lahtonen | ||
| S Oct 13, 2021 at 13:40 | history | bounty started | Jyrki Lahtonen | ||
| S Oct 13, 2021 at 13:40 | history | notice added | Jyrki Lahtonen | Draw attention | |
| Sep 11, 2021 at 11:18 | answer | added | orangeskid | timeline score: 4 | |
| S Mar 2, 2021 at 18:53 | history | bounty ended | Jyrki Lahtonen | ||
| S Mar 2, 2021 at 18:53 | history | notice removed | Jyrki Lahtonen | ||
| Mar 2, 2021 at 11:00 | answer | added | Thomas Rot | timeline score: 5 | |
| Mar 1, 2021 at 14:38 | history | edited | C.F.G |
tag added
|
|
| Mar 1, 2021 at 10:03 | answer | added | C.F.G | timeline score: 11 | |
| Mar 1, 2021 at 9:57 | comment | added | Jyrki Lahtonen | @C.F.G I believe you. The history of this question is complicated, and we didn't remember Morse theory at the beginning (and apparently it is not as widely known as it deserves to be). An explanation along those lines is a welcome answer! | |
| Mar 1, 2021 at 9:53 | comment | added | Jyrki Lahtonen | @C.F.G Yes, that is clear. Right now the question is Under what circumstances will the presence of two global minima imply the existence of a saddle point? (at the level the two lakes merge) Feel free to add extra assumptions (only isolated criticial points, compact domain or function tending to infinity or...) | |
| Mar 1, 2021 at 9:47 | comment | added | C.F.G | @JyrkiLahtonen: I don't understand something. all continuous functions on a compact domain attaint a max an a min at least. So what is the wrong with sphere or torus? | |
| S Feb 23, 2021 at 5:57 | history | bounty started | Jyrki Lahtonen | ||
| S Feb 23, 2021 at 5:57 | history | notice added | Jyrki Lahtonen | Draw attention | |
| S Feb 22, 2021 at 19:16 | history | bounty ended | Jyrki Lahtonen | ||
| S Feb 22, 2021 at 19:16 | history | notice removed | Jyrki Lahtonen | ||
| Feb 22, 2021 at 19:13 | history | edited | Jyrki Lahtonen | CC BY-SA 4.0 |
deleted 271 characters in body
|
| Feb 20, 2021 at 15:57 | comment | added | Xander Henderson♦ | In any event, that might be a productive direction to look for a more analytic proof. | |
| Feb 20, 2021 at 15:57 | comment | added | Xander Henderson♦ | I finally remembered yesterday why this problem felt familiar---it is something which is addressed by Morse theory. In particular, this seems to be related to Reeb's Theorem: roughly, a compact smooth manifold with exactly two nondegenerate critical points is homeomorphic to a sphere. There are some issues here---$\mathbb{R}^2$ is not compact, and user21820's observation that the result holds for functions which diverge to infinity suggests that this is a result about manifolds with boundaries. | |
| Feb 20, 2021 at 4:59 | answer | added | River Li | timeline score: 14 | |
| Feb 18, 2021 at 0:00 | history | tweeted | twitter.com/StackMath/status/1362190197341962242 | ||
| Feb 17, 2021 at 16:11 | answer | added | user21820 | timeline score: 32 | |
| Feb 17, 2021 at 5:54 | history | edited | Jyrki Lahtonen | CC BY-SA 4.0 |
added 1 character in body
|
| Feb 17, 2021 at 5:53 | comment | added | Jyrki Lahtonen | @EricTowers It could be interesting to see how the argument using Poincaré-Hopf you had in mind relates to Martin's counterexample. And also whether it leads to an affirmative results (which?) on another manifold (other than the uninteresting observation that on a compact manifold we also achieve a maximum). | |
| Feb 16, 2021 at 19:42 | comment | added | Jyrki Lahtonen | A few of us already discussed this question in the Pearl Dive. I don't usually get this involved in anything resembling calculus, but I want to diversify the Pearl Dive a bit (or at least the role I have there). | |
| S Feb 16, 2021 at 19:38 | history | bounty started | Jyrki Lahtonen | ||
| S Feb 16, 2021 at 19:38 | history | notice added | Jyrki Lahtonen | Reward existing answer | |
| Feb 14, 2021 at 15:31 | comment | added | Martin R | @JyrkiLahtonen: Maybe, I have no idea right now. | |
| Feb 14, 2021 at 15:17 | history | edited | Jyrki Lahtonen | CC BY-SA 4.0 |
added 165 characters in body
|
| Feb 14, 2021 at 5:16 | comment | added | Jyrki Lahtonen | @MartinR The counterexample does show a way of avoiding saddle points. May be a variant like: Compact domain + only isolated critical points (yours had "a ring of critical points along the equator") + two global minima => a saddle point? | |
| Feb 14, 2021 at 4:40 | history | became hot network question | |||
| Feb 13, 2021 at 21:31 | comment | added | Martin R | @JyrkiLahtonen: ... which made me realize that my “counterexample” on $S^2$ must be wrong. | |
| Feb 13, 2021 at 21:23 | comment | added | Jyrki Lahtonen | Wait! Don't we also have a maximum on a compact manifold? I am open to suggestions for better variants :-) | |
| Feb 13, 2021 at 21:09 | comment | added | Jyrki Lahtonen | @EricTowers Do you think differential-topology would be an appropriate tag? | |
| Feb 13, 2021 at 21:08 | answer | added | Martin R | timeline score: 88 | |
| Feb 13, 2021 at 21:06 | history | edited | Jyrki Lahtonen | CC BY-SA 4.0 |
added 63 characters in body
|
| Feb 13, 2021 at 20:58 | comment | added | Jyrki Lahtonen | Probably not @EricTowers. May be Poincaré-Hopf is not that well known among all and sundry :-) A number of us discussed the question without reaching a conclusion! | |
| Feb 13, 2021 at 20:56 | history | edited | Jyrki Lahtonen | CC BY-SA 4.0 |
edited title
|
| Feb 13, 2021 at 20:56 | comment | added | Eric Towers | Am I being slow? This seems like a direct application of Poincare-Hopf to the gradient of $f$. | |
| Feb 13, 2021 at 20:50 | history | edited | Jyrki Lahtonen | CC BY-SA 4.0 |
deleted 3 characters in body
|
| Feb 13, 2021 at 20:39 | history | asked | Jyrki Lahtonen | CC BY-SA 4.0 |