45

Here is a prime example of a MO-Hard question that doesn't appear on the unanswered list.
Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$?
The title says it all. In fact, the body of the question is just one line. It has had 16 answer attempts though most have been deleted after an error was found...

43

Mark Sapir solved an interesting open problem in group theory in this answer to a question of Narutaka OZAWA.

36

Our article D. Brumleve, J. D. Hamkins, and P. Schlicht, “The mate-in-n problem of infinite chess is decidable,” LNCS 7318(2012):78-88 was inspired directly by Richard Stanley's question Decidability of chess on an infinite board.
Abstract. Infinite chess is chess played on an infinite edgeless chessboard. The familiar chess pieces move about according to ...

33

A meta-answer: http://search.arxiv.org:8081/?query=mathoverflow&in= returns a list of 197 papers on the arXiv which mention MathOverflow.
Nearly all of these are actual citations, with a small number of papers about MathOverflow itself, and some number of papers which mention MathOverflow without giving full attribution according to the guidelines.
...

32

In April 2013 I discussed with Alan Sokal the following conjecture: if $P$ is a real polynomial with the property $|P(z)|<P(|z|)$ then some power of $P$ has positive coefficients.
We did not prove it at that time.
In August, Ofer Zeitouni asked on MO to describe all possible limits of the so-called empirical measures of polynomials with positive ...

30

This question is a very nice topological one, a cousin of Brouwer's fixed point theorem and related to several questions in the literature:
Suppose that $f$ and $g$ are two commuting continuous mappings from the closed unit disk to itself. Does there always exist a point $x$ such that $f(x)=g(x)$?

29

If you have an excuse to mention MO in your abstract, please consider doing so. MO is mentioned in the paper I submitted for the proceedings and it will appear in one slide of my talk at the ICM.

27

This is an old and self-indulgent story; but it was such a charmingly unexpected bonus from my early use of MathOverflow, that I think it deserves to be recorded somewhere (my apologies for its length!):
tl;dr: As a serendipitous consequence of this MathOverflow question, the second answerer invited me to give my first-ever seminar talk as a grad student.
...

26

Ben Green's paper on (not) computing the Möbius function arose from this question on MathOverflow.
Abstract. Any function $F : \{1,\dots,N\} \rightarrow \{-1,1\}$ such that $F(x)$ can be computed from the binary digits of $x$ using a bounded depth circuit is orthogonal to the Möbius function $\mu$ in the sense that $\frac{1}{N} \sum_{x \leq N} \mu(x)F(x) = ...

25

Tom Church, Melody Chan, and Joshua Grochow just posted their paper "Rotor-routing and spanning trees on planar graphs" to the arXiv here. It answers this MO question which was asked by Jordan Ellenberg.

25

The MO question
Norms of Commutators
is referenced in a paper by Ozawa, Schechtman and me that will appear in the PNAS. arXiv:1202.0986

23

Our article, C. D. A. Evans and J. D. Hamkins, Transfinite game values in infinite chess, where we investigate the range of transfinite game values arising in infinite chess, grew directly out of Johan Wästlund's question Checkmate in $\omega$ moves?.
In particular, we define the omega one of chess $\omega_1^{\frak{Ch}}$ to be the supremum of the ordinal ...

20

The paper A Counterexample to a Conjecture of Schwartz by Brandt, Chudnovsky, Kim, Liu, Norin, Scott, Seymour, and Thomassé answers this MO question of Felix Brandt. The question asks whether a weakened form of Schwartz’ Conjecture (a popular conjecture in Social Choice Theory) is true. The paper proves that even this weakened form of the conjecture is ...

19

This paper (details below) by Zhen Lin Low and Aaron Mazel-Gee cites not just MO but:
This collaboration would not have happened without the ‘Homotopy Theory’
chat room on MathOverflow.
arXiv.org > math > arXiv:1409.8192
From fractions to complete Segal spaces
Zhen Lin Low, Aaron Mazel-Gee
We show that the Rezk classification diagram of a relative ...

19

I make no claim to the significance of any of these problems,
nor that they are even "hard"—some are too peripheral to attract interest.
With those caveats, here are a few of mine (in chronological order):
"Which convex bodies roll along closed geodesics?" (Apr 2011): An Easter posting
:-) Some examples provided by Robert Bryant.
"Homometric ⇒ ...

19

Update: Donations are now possible, see this post: Donations to MathOverflow, Inc
[Apologies for the long delay in answering, I was on my honeymoon! I may update this quick answer after I get back to regular life.]
Up until now, MathOverflow has been paying for regular expenses through a generous grant from The Alfred P. Sloan Foundation. Coincidentally, ...

18

I posted a question about a year ago on MO (and even further back on MSE) that remains elusive:
Randomly break a stick in five places.
Question: What is the probability that the resulting six pieces can form a tetrahedron?
The two breaks to form a triangle problem has a collection of proofs on MO here; the five breaks for a tetrahedron problem is ...

16

The Opinion column "Mathematical Community" in the March 2011 issue, by John Swallow, asks, "Are mathematicians at the forefront of collaboration, with the advent of the Polymath Projects and Math Overflow?"
There's a passing mention in the January 2013 issue in "When 7,000 Mathematicians Come to Boston" by Alexi Hoeft of a discussion session at the JMM "...

15

This MO question was asked in December of 2011, in line with a reference request for a senior thesis on odd perfect numbers completed in 1978. Subsequently, the OP has tried numerous ways to get hold of the thesis's author.
On August 24, 2013 Jim (Condict) Grace (the thesis's author) popped in to MO to respond to the original question.

15

It seems to me that one of the principal ways to attract serious people in a given research area is to have an abundance of interesting, sophisticated questions in that area. This is particularly true if such questions do not get readily answered.
This suggests that it may suffice to have the cart before the horse, so to speak, in that one might attract ...

15

The swag is here!
The re-order shipment is now in stock and that information we've been holding for over a year now has been sent off to our warehouse so the orders can now be processed. Everything will be shipped shortly and should start arriving at your doorsteps (or wherever you get your mail) over the coming weeks. Enjoy!

15

Simon Thomas asked in
Ultrafilters and automorphisms of the complex field
whether the existence of non-principal ultrafilters (over the natural numbers) suffices to imply the existence of a nontrivial automorphism of the complex field $\mathbb C$. In set theoretic terms, the question is whether (under appropriate large cardinal assumptions) there is such ...

14

I am frequently asked whether exponentials are "linearly independent". That is if we have
a sequence of distinct complex numbers $\lambda_j$, whether
$$\sum_{j} a_j\exp(\lambda_j z)\equiv 0$$
implies that all $a_j=0$. If the linear dependence above holds in all complex plane, this
question was answered by A. F. Leontjev. All his books exist only in Russian, ...

14

My joint article with Justin Palumbo, The rigid relation principle, a new weak choice principle (Mathematical Logic Quarterly 58(6):394-398, 2012) grew out of our answers to my question, Does every set admit a rigid binary relation? (and how is this related to the Axiom of Choice?), which grew out of Mike Shulman's question, A rigid type of structure that ...

14

My MO question Conjugation of group extensions was answered by Ycor. As a result, we wrote a joint note
Conjugate complex homogeneous spaces with non-isomorphic fundamental groups published in C. R. Acad. Sci. Paris, Ser. I, 353 (2015) 1001–1005.

14

The question whether there is a non surjective bounded linear operator on $\ell_\infty$ that has dense range was answered in this paper by Amir Bahman Nasseri, Gideon Schechtman, Tomasz Tkocz, and me. An interesting aspect of the proof is that it uses a theorem proved by computer scientists to get a counterexample. So, in some sense, this question about ...

14

One I like very much is
Can we color $\mathbb Z^+$ with $n$ colors such that $a, 2a, \dots, na$ all have different colors for all $a$?
Among the failed proposed attempts, the ones that appeared most promising were in essence group- or number-theoretic, for instance the question of whether the partial graph of multiplication $$\{(a,b,c)\in\{1,\dots,n\}^3\...

14

A question that fascinates me is the problem of characterizing the subsets $S$ of the unit circle $\mathbb T$ for which there is a power series with radius of convergence $1$ that, on $\mathbb T$, converges on $S$ and diverges otherwise.
Behaviour of power series on their circle of convergence.
(The question has also been asked on MSE a couple of times.)

13

The analog of the famous law of iterated logarithm for maximum eigenvalue of a random Gaussian matrix was asked here. Zeitouni's MO-answer was expanded (after significant effort) to a full answer for the limsup (including constants) and a partial answer for the liminf by Elliot Paquette and Ofer Zeitouni arxiv.org/abs/1505.05627 !

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