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.
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.
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.
An unpublished open problem posed by Adam Chalcraft, Does every polyomino tile $\mathbb R^n$ for some $n$?, received considerable attention when I posted it here on MO.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
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.
In the MO question Normalizers in symmetric groups I had asked the following:
Let $G$ be a finite group. Is it true that there is a subgroup $U$ inside some symmetric group $S_n$, such that $N(U)/U$ is isomorphic to $G$? Here $N(U)$ is the normalizer of $U$ in $S_n$.
This question received quite a few upvotes. Still even offering a bounty some years ago left it open.
Now, 12 years later, Alexei Entin and Cindy Tsang gave a positive answer to this question in their preprint Normalizer Quotients of Symmetric Groups and Inner Holomorphs. One of the main tools is Yves de Cornulier's answer to the MO question Is every finite group the outer automorphism group of a finite group? and its subsequent variant by Benjamin Sambale.
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 $d \le 4$.
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.
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.''
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.
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.
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).
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.
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.
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".
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".
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."
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.
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.
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.
Leonardo Zapponi has answered my question Parametric solutions of Pell's equation in his paper Parametric solutions of Pell equations.
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 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!!!
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).
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.
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
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
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.
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.
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. :)
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:
and then again in this paper, as Lemma 2.4:
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
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.