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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.

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    $\begingroup$ This question was suggested in this discussion. $\endgroup$ Aug 12, 2013 at 15:07
  • $\begingroup$ 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. $\endgroup$ Aug 12, 2013 at 18:22
  • $\begingroup$ @David: Clarify what you're suggesting. What do you propose migrating this way? $\endgroup$ Aug 12, 2013 at 18:27
  • $\begingroup$ 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. $\endgroup$ Aug 12, 2013 at 19:01
  • $\begingroup$ @David: The thread at tea is mostly other stuff so it's best to do it manually. $\endgroup$ Aug 12, 2013 at 19:10
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    $\begingroup$ @DavidWhite tangentially, but one could add things in the old thread. $\endgroup$
    – user9072
    Aug 12, 2013 at 21:15
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    $\begingroup$ 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. $\endgroup$
    – qed
    Aug 24, 2013 at 16:09
  • $\begingroup$ @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. $\endgroup$ Dec 26, 2016 at 15:02
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    $\begingroup$ 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. $\endgroup$ Apr 9, 2017 at 1:57
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    $\begingroup$ @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 $\endgroup$ Jan 21, 2018 at 9:49
  • $\begingroup$ This is sort of a duplicate of this closed question: mathoverflow.net/q/11846/1345 $\endgroup$
    – Ian Agol
    Nov 19, 2019 at 18:41
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    $\begingroup$ @Ian I think the idea was that meta is a more appropriate place for this question than main was. $\endgroup$ Dec 3, 2019 at 22:37
  • $\begingroup$ As the link to tea given at the end of the post does not work now, I'll add a link to Wayback Machine at least here in a comment. $\endgroup$ Jun 14, 2022 at 5:31
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    $\begingroup$ I'd like to shorten the current title "Best of MathOverflow, or papers inspired by MathOverflow" to "Research inspired by MathOverflow". Two reasons: one is that I'd feel uncomfortable in claiming a paper (especially mine) as "best of". Second, it just better suits the question. (In practice it's rather "Papers inspired by MO", but the question allows a broader interpretation.) What do you think? $\endgroup$
    – YCor
    Jul 10, 2022 at 14:33

60 Answers 60

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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.

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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.

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An unpublished open problem posed by Adam Chalcraft, Does every polyomino tile $\mathbb R^n$ for some $n$?, received considerable attention when I posted it here on MO. (Of all the questions that I have posed on MO that aren't soft questions, it has received the most upvotes—69 as of this writing.) It was solved by Vytautas Gruslys, Imre Leader, and Ta Sheng Tan, who learned about it from MO, in

Vytautas, Gruslys, Imre Leader, Ta Sheng Tan, Tiling with arbitrary tiles, Proc. Lond. Math. Soc. (3) 112 No. 6 (2016) 1019–1039, doi:10.1112/plms/pdw017, arXiv:1505.03697

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    $\begingroup$ And just to close the loop: They provided a positive answer. $\endgroup$ Dec 5, 2019 at 0:26
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The question, "How hard is reconstructing a permutation from its differences sequence?" posed by Mohammad Al-Turkistany, was answered by Marzio De Biasi, who then wrote a paper, "Permutation Reconstruction from Differences," published in the Electronic Journal of Combinatorics (2014):

We prove that the problem of reconstructing a permutation $\pi_1,\ldots,\pi_n$ of the integers $[1\ldots n]$ given the absolute differences $|\pi_{i+1}-\pi_i|$, $i=1,\ldots,n−1$, is 𝖭𝖯-complete.

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This paper,

Roman Karasev, Jan Kynčl, Pavel Paták, Zuzana Safernová, and Martin Tancer. "Bounds for Pach's selection theorem and for the minimum solid angle in a simplex." arXiv:1403.8147 (2014). Discrete & Computational Geometry. 54:610-636 (2015).

cites my answer to Boris Bukh's question, Angle of a regular simplex, in the discussion of their theorem giving an upper bound on the minimum solid angle of a $d$-simplex.

Incidentally, they pose a very nice question:

Is it true that the minimum solid angle of a $d$-simplex is at most the solid angle of a regular $d$-simplex?

The answer is Yes for $d \le 4$.

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  • $\begingroup$ In context, they write "This shows that our bound in Theorem 6 is tight up to lower order terms in the exponent. Rogers’ proof is also reproduced in a book by Zong [Zon99, Lemma 7.2]. We have learnt about this from an answer of Joseph O’Rourke [O’R11] to a question of Boris Bukh at MathOverflow." $\endgroup$ Jan 19 at 2:38
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Julien Marché's question "Homology generated by lifts of simple curves" was the first appearance in print of a folklore question (I first was asked it back when I was a postdoc). As I discuss in my answer here, there have been a number of recent papers addressing it, including

J. Malestein, A. Putman, Simple closed curves, finite covers of surfaces, and power subgroups of $\text{Out}(F_n)$, preprint 2017.

T. Koberda, R. Santharoubane, Quotients of surface groups and homology of finite covers via quantum representations, Invent. Math. 206 (2016), no. 2, 269–292.

B. Farb, S. Hensel, Finite covers of graphs, their primitive homology, and representation theory, New York J. Math. 22 (2016), 1365–1391.

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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.''

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My Forum of Mathematics Sigma paper (published 2021) answered a 20-year old question of Jeff Shallit. The proof makes crucial use of ideas in a 2016 MO answer by Anthony Quas.

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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.

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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.

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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.

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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".

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    $\begingroup$ American spelling for MO, British for arXiv? $\endgroup$ May 15, 2019 at 13:32
  • $\begingroup$ That's right, we had to upgrade the spelling from MO to arXiv :) $\endgroup$ Aug 23, 2022 at 13:26
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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."

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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.

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  • $\begingroup$ 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. $\endgroup$ Dec 3, 2019 at 17:29
  • $\begingroup$ @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? $\endgroup$ Dec 3, 2019 at 18:37
  • $\begingroup$ 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. $\endgroup$ Dec 3, 2019 at 18:52
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    $\begingroup$ @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. $\endgroup$ Dec 3, 2019 at 19:00
  • $\begingroup$ 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. $\endgroup$ Dec 3, 2019 at 19:14
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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.

MattParker

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    $\begingroup$ Such a fun video! $\endgroup$ May 23, 2021 at 0:56
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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.

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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.

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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!!!

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    $\begingroup$ push? publish?? $\endgroup$ May 6, 2019 at 4:11
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    $\begingroup$ I think it wouldn't hurt to edit in links to whatever you have written up (if your results are ready for publication). $\endgroup$ May 6, 2019 at 6:57
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    $\begingroup$ 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... $\endgroup$
    – Jon Bannon
    May 6, 2019 at 7:19
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Garabed Gulbenkian's 2010 question about ordinal definable real numbers, which Joel David Hamkins called "a fascinating, outstanding question," attracted considerable attention, culminating in a 2015 paper by Kanovei and Lyubetsky, A definable ${\mathsf E}_0$ class containing no definable elements. The paper credits MathOverflow as the source of the question (although, curiously, Gulbenkian is not mentioned by name).

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I just learned that a question of mine on minimal vertex covers in hypergraphs where edges intersect in at most $1$ point led to a paper by Lajos Soukup and Tamas Csernak; it will appear in Discrete Mathematics in 2023. The paper does not fully answer the original question - but congratulations are in order to Lajos and Tamas for their progress.

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Zhi-Wei Sun posted a highly upvoted MO question in 2018, Can we write each positive rational number as ${1\over p_1−1}+\cdots+{1\over p_k−1}$ with $p_1,\ldots,p_k$ distinct primes? The question remained without an answer until May 2023, when Thomas Bloom posted an answer to MO and a preprint to the arXiv (though the preprint does not mention MathOverflow).

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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".

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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

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Beginning with discussions in MO, it was proved that the Apéry sequence is a Stieltjes moment sequence. My final streamlined argument for this was posted to the arxiv in 2020: LINK

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In response to an MO question, On maximal regular polyhedra inscribed in a regular polyhedron, Moritz Firsching filled in the missing cases from H.T. Crofts original paper on the topic.

He then posted a paper to the arXiv in 2014, "Computing maximal copies of polytopes contained in a polytope." arXiv abstract, and it was published in 2015 in Experimental Mathematics Vol. 24 (2015), Issue 1, pp.98-105.

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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.

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

From the answer:

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. :)

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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.

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

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