Timeline for A long-lasting conjecture of Pólya & Szegő
Current License: CC BY-SA 4.0
22 events
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| Jun 13 at 5:54 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
fixed a minor typo + included the title of the linked paper
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| May 29 at 10:48 | answer | added | Beni Bogosel | timeline score: 7 | |
| Feb 4, 2019 at 15:47 | vote | accept | BigM | ||
| Feb 23, 2017 at 0:12 | comment | added | BigM | @Suvrit I perhaps overlooked at your comment when you posted it. Selberg's conjecture is definitely relevant. However it is not clear how domains being polygon play any rule. | |
| Nov 14, 2016 at 18:32 | history | edited | BigM | CC BY-SA 3.0 |
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| Nov 14, 2016 at 2:24 | history | edited | BigM | CC BY-SA 3.0 |
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| Nov 14, 2016 at 1:25 | history | edited | BigM | CC BY-SA 3.0 |
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| May 3, 2016 at 21:33 | comment | added | BigM | @BeniBogosel Well not really, but I actually worked on an equivalent problem for logarithmic and Newtonian integral operators. I have an idea using symmetries of operator norms which might be a new approach. see my paper and Ruzhansky's paper.Basically the largest eigenvalue of certain operator with log. kernel is equal to the least eigenvalue for the Laplacian, and working with eigenvalues of the mentioned operator is much simpler (in my opinion). | |
| May 3, 2016 at 21:17 | comment | added | Beni Bogosel | @BigM: Did you make some progress on the conjecture? A published result? | |
| May 2, 2016 at 17:50 | comment | added | BigM | @BeniBogosel very interesting. I actually had seen your page a long time ago while actively thinking about Polya's conjecture. | |
| Apr 30, 2016 at 21:43 | comment | added | Beni Bogosel | Some recent numerical computations suggest that regular polygons are indeed minimizers of the first eigenvalue under area constraint: lama.univ-savoie.fr/~bogosel/faber_krahn_polygons.html I do not know any proof of the fact that regular polygons are local minimizers. It is possible to prove that they are critical points, i.e. the derivative of the first eigenvalue of a regular polygon is zero with respect to every perturbation of the vertices. | |
| Apr 10, 2016 at 20:59 | comment | added | BigM | At Survit, I looked at the link(Peter Sarnak's) paper on Selberg's conjecture.That is not in my realm however to my understanding Selberg's inequality is analog of general isopremeteric inequality stating among regions of same area circle(balls for higher dimensions)maximizes the first eigenvalue. M.Ruzhansky(Imperial College London) has some nice results pertaining iso. inequalities for Lie groups and connected Riemannian manifolds. | |
| Apr 10, 2016 at 16:48 | comment | added | Suvrit | Is this conjecture also connected in any way to the Selberg eigenvalue conjecture, or they just happen to share the word "eigenvalue"? | |
| Apr 10, 2016 at 16:08 | comment | added | BigM | @NoamD.Elkies As far as I know, the Polya's proof is the only one out there. I have not seen any proof for even "local"minimality. I thought I had a good idea how to approach this problem but sadly my method is only good for quadrilaterals. | |
| Apr 10, 2016 at 15:32 | answer | added | Otis Chodosh | timeline score: 15 | |
| Apr 10, 2016 at 15:19 | comment | added | Noam D. Elkies | Is the regular $n$-gon even proved to be a local minimum for $\lambda_1$? The survey you cite doesn't cite such a result in the "case of polygons" sections (3.2, page 5). | |
| Apr 10, 2016 at 2:52 | history | edited | GH from MO | CC BY-SA 3.0 |
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| Apr 10, 2016 at 1:54 | history | edited | GH from MO | CC BY-SA 3.0 |
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| Apr 10, 2016 at 1:50 | history | edited | Yoav Kallus |
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| Apr 10, 2016 at 1:49 | history | edited | BigM | CC BY-SA 3.0 |
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| Apr 10, 2016 at 1:42 | history | edited | BigM | CC BY-SA 3.0 |
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| Apr 10, 2016 at 1:36 | history | asked | BigM | CC BY-SA 3.0 |