# Best of MathOverflow, or papers inspired by MathOverflow

This is a place to collect MathOverflow success stories!

Was some of your research inspired by something on MathOverflow? Do you know questions & answers that led to interesting research? MathOverflow citations? Open problems solved on MathOverflow? Then add your story in an answer! (One story per answer, please!)

If you want to help get this thread started, you can use this search to find MathOverflow citations on the arXiv or migrate some old stories from tea.mathoverflow.net.

• This question was suggested in this discussion. – François G. Dorais Aug 12 '13 at 15:07
• Is it really the case that the old success stories have to be migrated by hand? I'm surprised there's not a way to do this within the stack software or at least write a program to do it. – David White Aug 12 '13 at 18:22
• @David: Clarify what you're suggesting. What do you propose migrating this way? – François G. Dorais Aug 12 '13 at 18:27
• Hi. I felt that if we plan to have a single unified place for success stories (e.g. if someone wants to come and write an article about MO) then it doesn't make sense to have one place at tea for pre-migration and one here for post-migration. So the old success stories should be here, since it appears we can't add new ones to that old page. But it would take a long time to move them over one by one. I wish it could be done all at once. – David White Aug 12 '13 at 19:01
• @David: The thread at tea is mostly other stuff so it's best to do it manually. – François G. Dorais Aug 12 '13 at 19:10
• @DavidWhite tangentially, but one could add things in the old thread. – user9072 Aug 12 '13 at 21:15
• Many papers were inspired by questions raised on MO, but has any credit been given to those who raised the questions in the publications? I am just wondering. – qed Aug 24 '13 at 16:09
• @qed I know for a fact that in many instances, perhaps most, the MO question-asker is given credit and citations. Further, I know of several cases, and I expect that there are many more, in which the question-asker was invited to join as co-author. – Joel David Hamkins Dec 26 '16 at 15:02
• I wonder if the title should be changed from "Best of MathOverflow" to "Publications resulting from MathOverflow" (or maybe "inspired by"?). It has in fact developed into publication citing. – Joseph O'Rourke Apr 9 '17 at 1:57
• @JosephO'Rourke What you write is perhaps true. OTOH the ones which MO community considers the best are quite likely to rise to the top based on voting, so in this sense it might still be a fitting title. I'll point out that Todd Trimble recently commented on the title of this thread: Some people might feel uneasy citing their own work at a thread entitled "Best of MathOverflow", but perhaps that title should be interpreted broadly – Martin Sleziak Jan 21 '18 at 9:49
• This is sort of a duplicate of this closed question: mathoverflow.net/q/11846/1345 – Ian Agol Nov 19 '19 at 18:41
• @Ian I think the idea was that meta is a more appropriate place for this question than main was. – Gerry Myerson Dec 3 '19 at 22:37

Stefan Kiefer and Björn Wachter just published a paper, "Stability and Complexity of Minimising Probabilistic Automata" (arXiv link), which acknowledges the MO question convex polyhedron in the unit cube.

The MO question, "Shortest closed curve to inspect a sphere," was cited as the "initial stimulus" for the paper

He establishes a lowerbound of $6\sqrt{3}$ on the shortest inspection curve, more than $80$% of the conjectured $4 \pi$ lowerbound.

Mohammad Ghomi answered the question Shortest closed curve to inspect a sphere, in a paper, Shortest closed curve to inspect a sphere, posted to the arXiv (https://arxiv.org/abs/2010.15204), whose PDF is available here, verifying the conjecture that the shortest curve is four consecutive semicircles each of length $$\pi$$ for a unit sphere.

The answer to the question Length of Hirzebruch continued fractions was published as a short note On continued fractions of equal length .

Yoav Kallus gave interesting improvement in “The Two Sheriffs” puzzle. It is not a serious open problem but he gave a really surprizing answer using Fano plane.

The paper "Majority colourings of digraphs" by Paul Seymour, Stephan Kreutzer, Sang-il Oum, David R. Wood and myself has its origin in my question "Majority coloring for directed graphs".

• American spelling for MO, British for arXiv? – Gerry Myerson May 15 '19 at 13:32

The full answer to the question Decidability of diophantine equation in a theory by rainmaker in the case of Robinson’s arithmetic was written up in my paper Division by zero, Archive for Mathematical Logic 55 (2016), no. 7, pp. 997–1013.

This doesn't quite fit the mold of the other postings, but Matt Parker (Numberphile and StandUpMaths) made a YouTube video that mentions MathOverflow several times, and particularly highlights the work of Moritz Firsching.

• Such a fun video! – Sam Hopkins May 23 at 0:56

The paper, "A quantitative obstruction to collapsing surfaces," by Mikahil G. Katz, arXiv abs, addresses the MO question, "Gromov-Hausdorff limits of 2-dimensional Riemannian surfaces" posed by sva (S. Alesker).

Abstract. We provide a quantitative obstruction to collapsing surfaces of genus at least $2$ under a lower curvature bound and an upper diameter bound.

This paper of mine (Arithmetic Restrictions on Geometric Monodromy) was inspired in large part by this question asked by Lisa S., though the original motivation is not so obvious in the final product.

I asked a somewhat silly question, which silliness was pointed out by Tobias Fritz, which answer I cited just to be pedantic about a point in set theory in a paper I wrote recently.

Shengkui Ye, in the review in Math Reviews of Bela Bauer and Claire Levaillant, A new set of generators and a physical interpretation for the SU(3) finite subgroup D(9,1,1;2,1,1), Quantum Inf. Process. 12 (2013), no. 7, 2509–2521, MR3065503, cites the discussion at The finite subgroups of SU(n) as contradicting the claim by Bauer and Levaillant that "After 100 years of effort, the classification of all the finite subgroups of SU(3) is yet incomplete.''

Joachim König has answered my question Order of products of elements in symmetric groups in his paper A note on the product of two permutations of prescribed orders, to appear in European Journal of Combinatorics.

Leonardo Zapponi has answered my question Parametric solutions of Pell's equation in his paper Parametric solutions of Pell equations.

The discussion initiated by my question Primes occurring as orders of elements of a finitely presented group led to the addition of Section 5 to:

Maurice Chiodo, On torsion in finitely presented groups, Groups Complexity Cryptology 6(1): 1-8 (2014). arXiv version.

Quoting Oscar Cunningham's answer to an MO question asking for decision problems that are not known to be decidable:

In Conway's Game of Life, the problem of deciding whether a given pattern with finitely many live cells is a Garden of Eden (i.e. whether it lacks a predecessor).

Added 2019 December 3: Having learnt about the problem from this post, Ville Salo and Ilkka Törmä have produced a paper (Gardens of Eden in the Game of Life) showing that this problem is decidable.

• You might review the post, as some part of the problem is open, and your excerpt does not reflect that. Gerhard "Let's Not Do False Advertising" Paseman, 2019.12.03. – Gerhard Paseman Dec 3 '19 at 17:29
• @GerhardPaseman : What I've written here looks correct to me. I intentionally omitted the extra open problem that Oscar Cunningham introduced, as a potential road to proving the main open problem that was ultimately not used by Salo and Törmä. Do you see something wrong? – Timothy Chow Dec 3 '19 at 18:37
• Yes, but it is probably my misunderstanding (or misperception) of the problem. If you exercised your customary due diligence, then I don't think you need to change it. My impression was that they (the authors of the paper) had only solved a decision problem of whether a finite configuration had a finite predecessor, not the decision problem of whether it had a predecessor. Gerhard "Maybe I'm Mixing Up Quantifiers?" Paseman, 2019.12.03. – Gerhard Paseman Dec 3 '19 at 18:52
• @GerhardPaseman : Let C1 = "there exists an algorithm for deciding whether a given pattern with finitely many live cells is a Garden of Eden" and let C2 = "there is no non-Garden-of-Eden with finitely many live cells all of whose predecessors have infinitely many live cells." C1 is the main question of interest, which is now proved. Cunningham pointed out that C2 implies C1, but Salo and Törmä's proof of C1 did not go by way of C2, which remains open. – Timothy Chow Dec 3 '19 at 19:00
• Ah. For the careful reader, there may be no need to mention C2. For the casual reader (which seems to be me more often lately, but might also be students reading your post), your comment about C2 adds much to understanding. I leave it to you whether to include the comment material in the post above. I strongly recommend leaving your comment for future readers. Gerhard "Learns Much From Reading Comments" Paseman, 2019.12.03. – Gerhard Paseman Dec 3 '19 at 19:14

This MO question "Property $$\Gamma$$ in terms of correspondences" led us to answer two old open problems and to push further a third more recent result:

Jon Bannon, Amine Marrakchi, Narutaka Ozawa. Full factors and co-amenable inclusions, arxiv/1903.05395.

Thanks MO!!!

• push? publish?? – Gerry Myerson May 6 '19 at 4:11
• I think it wouldn't hurt to edit in links to whatever you have written up (if your results are ready for publication). – Gerry Myerson May 6 '19 at 6:57
• I see. Taka linked to the paper in the comment following his answer, which set this in motion. I'll link to the paper in the answer here... – Jon Bannon May 6 '19 at 7:19

This paper addresses and partially solves a question posed by Matthew Kahle, whose MO question they explicitly cite: chromatic number of the hyperbolic plane.

DeCorte, Evan, and Konstantin Golubev. "Lower bounds for the measurable chromatic number of the hyperbolic plane." Discrete & Computational Geometry 62, no. 2 (2019): 481-496. Journal link.

"Using spectral methods, we prove that if the colour classes are measurable, then at least six colours are needed to properly colour $$\mathbb{H}(d)$$ when $$d$$ is sufficiently large."

Ilya Bogdanov has answered my question Graphs with only disjoint perfect matchings on certain coloring in graphs, that emerged through research in quantum physics. This answer has inspired quite a bit of research:

Furthermore in Questions on the Structure of Perfect Matchings inspired by Quantum Physics, we generalize the question on inherited colorings to cover the full potential of quantum physics. One of my co-authors is Daniel Soltész, who I only met through MO. In this paper we cited Bogdanov's MO answere again (and called it "Bogdanov's Lemma").

The Q1 of the article is another MO question"Vertex coloring inherited from perfect matchings (motivated by quantum physics)", but i have little hope that I get so lucky again.

This is certainly, by far, my personal "Best of MO".

Another candidate for this thread of Mathoverflow success stories: my recent answer to Finding the nearest matrix with real eigenvalues.

We started our research by thinking about this problem, but then we were happily surprised to find out that the technique can be applied also to a harder problem that had already been studied in the literature on numerical linear algebra and control theory, that of finding the nearest Hurwitz stable matrix. And, more generally, it can be extended to solve numerically the problem of finding $$B = \arg \min_{S_\Omega} \|B-A\|_F,$$ where $$S_\Omega$$ is the set of all (real or complex) matrices with all eigenvalues in a given closed set $$\Omega$$. Another nice example of how procrastinating and thinking about Mathoverflow questions can lead to useful research sometimes. :)

A paper by T. Chartier, P.P. Pach and myself describing the status of "Can we color Z^+ with n colors such that a, 2a, …, na all have different colors for all a?" has just been published.

Coloring the $$n$$-Smooth Numbers with $$n$$ Colors, The Electronic Journal of Combinatorics 28 (1) (2021), #P1.34. https://doi.org/10.37236/8492

Here are two kind-of dual results that have now been proved after I asked them here:

The first arose from Induced map on path manifolds: is it a submersion?, which got an answer in very general terms involving generalised smooth spaces in this preprint:

• Andrew Stacey, Yet More Smooth Mapping Spaces and Their Smoothly Local Properties, arXiv:1301.5493.

and then again in this paper, as Lemma 2.4:

• Habib Amiri, Alexander Schmeding A differentiable monoid of smooth maps on Lie groupoids, Journal of Lie Theory 29 (2019), No. 4, 1167–1192, arXiv:1706.04816

in a way that only partially overlaps with Andrew Stacey's version; in one sense it's less general, but it seems to use a stronger notion of submersion/regular map.

The second came from Extension of functions from geodesically convex compact sets in a Riemannian manifold, which has now been answered in

• David Michael Roberts, Alexander Schmeding, Extending Whitney's extension theorem: nonlinear function spaces, to appear, Annales de l'Institut Fourier, arXiv:1801.04126

A different kind of question I asked now has three papers giving three different approaches, namely On a weak choice principle, which led to (in chronological order):

• Benno van den Berg, WISC is independent of ZF, (pdf), also Theorem 5.1/Corollary 5.2 in Predicative toposes, arXiv:1207.0959. This uses Gitik's class forcing symmetric model of ZF, over ZFC with a reasonably strong large cardinal assumption.

• Asaf Karagila, Embedding Orders Into Cardinals With $$DC_\kappa$$, Fund. Math. 226 (2014), 143-156, doi:10.4064/fm226-2-4, arXiv:1212.4396. This uses class forcing symmetric models, over ZFC with no large cardinals.

• David Michael Roberts, The weak choice principle WISC may fail in the category of sets, Studia Logica Volume 103 (2015) Issue 5, pp 1005-1017, doi:10.1007/s11225-015-9603-6 arXiv:1311.3074. This uses topos-theoretic methods, over a well-pointed base topos with no Choice.

This preprint of Souvik Dey (on commutative ring theory) cites an MO post.