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## Mathematical research inspired in fundamental part by mathoverflow

Mathoverflow has led in several instances to new mathematical work that arose directly from mathematical ideas, questions or answers posted here. Articles containing such work, inspired in fundamental part by mathoverflow material, might be characterized as having been born on mathoverflow.

Let us collect together here references to such work. In each case, please include a link to the relevant mathoverflow post or posts, a link to the mathematical work, for example an article at the math arxiv or the relevant journal, and a very brief summary abstract. (More detailed abstracts would presumably be available for those following the links to the article.)

In order that this question will not become burdened with excessive posts, let's agree to the following principles:

1. Please makes posts here only about essentially completed articles, rather than works-in-progress.
2. Please make posts here only about articles whose main existence was inspired by mathoverflow, such as a case where the topic of a question or the essence of an answer became the main substance of an article.
3. Please do not make posts here concerning articles simply because they cite mathoverflow, or simply because mathoverflow was involved at a critical step of the article, since such citations will eventually become so numerous as to be unremarkable. Rather, make a post here only in connection with an article that could be characterized as essentially about the topic or ideas expressed on mathoverflow and the authors were directly inspired by that mathoverflow material.

Please note the related meta-thread for more general discussion about references to mathoverflow, and Gower's question on breakthroughs that seeks examples of situations where mathoverflow helped a researcher make a critical advance.

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I don't see why we cannot keep this on meta. – Thierry Zell Feb 15 2012 at 21:21
MO has a great impact on the mathematical community as a whole - highly disputable, and I speak as someone who likes the site. – Yemon Choi Feb 15 2012 at 21:31
It seems fine to me to have this list here on the main site, perhaps with the (modified) understanding that only completed papers, with links to journals or to the math arxiv, would be appropriate for posting. I would prefer it to the meta site discussion, in part because it will be nice to see it periodically pop up to the top, when someone completes a new MO-inspired paper. But with the modified understanding I proposed, this won't be so often as to become annoying. So I voted to re-open. – Joel David Hamkins Feb 16 2012 at 0:56
In particular, the meta-site forum for this information is rather discursive, and doesn't organize the information as well as MO question/answer format would. – Joel David Hamkins Feb 16 2012 at 0:59
Rather than having this comment thread get out of control, I recommend we move the discussion to meta. I've started a thread: meta.mathoverflow.net/discussion/1312/… Please upvote this comment so it appears "above the fold" – David White Feb 16 2012 at 20:10

The paper "The mate-in-n problem of infinite chess is decidable" by D. Brumleve, J. D. Hamkins, and me was inspired by a question asked on Mathoverflow. The paper is available on the arxiv. Please see J. D. Hamkins' blog for the abstract or if you would like to post a comment, here's a short version of the abstract:

"Infinite chess is chess played on an infinite edgeless chessboard. The familiar chess pieces move about according to their usual chess rules, and each player strives to place the opposing king into checkmate. The mate-in-n problem of infinite chess is the problem of determining whether a designated player can force a win from a given finite position in at most n moves. The main theorem of this article, confirming a conjecture of the second author and C. D. A. Evans, establishes that the mate-in-n problem of infinite chess is computably decidable, uniformly in the position and in n. The proof proceeds by showing that the mate-in-n problem is expressible in what we call the first-order structure of chess, which we prove (in the relevant fragment) is an automatic structure, whose theory is therefore decidable."

I'm looking forward to hearing about other papers inspired by Mathoverflow.

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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 $\frac{1}{N} \sum_{x \leq N} \mu(x)F(x) = o_{N \rightarrow \infty}(1)$. The proof combines a result of Linial, Mansour and Nisan with techniques of Kátai and Harman-Kátai, used in their work on finding primes with specified digits.

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 Thanks, Joel: I updated my post accordingly. – GH Mar 4 2012 at 20:58