Mathematical research inspired in fundamental part by mathoverflow - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T23:19:11Z http://mathoverflow.net/feeds/question/88563 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88563/mathematical-research-inspired-in-fundamental-part-by-mathoverflow Mathematical research inspired in fundamental part by mathoverflow Dirk 2012-02-15T21:18:49Z 2012-03-04T20:58:09Z <p>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.</p> <p>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.)</p> <p>In order that this question will not become burdened with excessive posts, let's agree to the following principles:</p> <ol> <li>Please makes posts here only about essentially completed articles, rather than works-in-progress.</li> <li>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.</li> <li>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.</li> </ol> <p>Please note the <a href="http://meta.mathoverflow.net/discussion/64/where-to-keep-track-of-math-overflow-success-stories/#Item_7" rel="nofollow">related meta-thread</a> for more general discussion about references to mathoverflow, and <a href="http://mathoverflow.net/questions/11846/has-mathoverflow-yet-led-to-mathematical-breakthroughs" rel="nofollow">Gower's question on breakthroughs</a> that seeks examples of situations where mathoverflow helped a researcher make a critical advance. </p> http://mathoverflow.net/questions/88563/mathematical-research-inspired-in-fundamental-part-by-mathoverflow/90217#90217 Answer by Philipp Schlicht for Mathematical research inspired in fundamental part by mathoverflow Philipp Schlicht 2012-03-04T18:48:12Z 2012-03-04T18:48:12Z <p>The paper "The mate-in-n problem of infinite chess is decidable" by D. Brumleve, J. D. Hamkins, and me was inspired by a <a href="http://mathoverflow.net/questions/27967" rel="nofollow">question asked on Mathoverflow</a>. The paper is <a href="http://arxiv.org/abs/1201.5597" rel="nofollow">available on the arxiv</a>. Please see <a href="http://jdh.hamkins.org/the-mate-in-n-problem-of-infinite-chess-is-undecidable" rel="nofollow">J. D. Hamkins' blog</a> for the abstract or if you would like to post a comment, here's a short version of the abstract: </p> <p>"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."</p> <p>I'm looking forward to hearing about other papers inspired by Mathoverflow. </p> http://mathoverflow.net/questions/88563/mathematical-research-inspired-in-fundamental-part-by-mathoverflow/90224#90224 Answer by GH for Mathematical research inspired in fundamental part by mathoverflow GH 2012-03-04T20:21:29Z 2012-03-04T20:58:09Z <p>Ben Green's <a href="http://front.math.ucdavis.edu/1103.4991" rel="nofollow">paper</a> on (not) computing the Möbius function arose from <a href="http://mathoverflow.net/questions/57543/walsh-fourier-transform-of-mobius-functions" rel="nofollow">this question</a> on MathOverflow.</p> <p><strong>Abstract.</strong> Any function <code>$F : \{1,\dots,N\} \rightarrow \{-1,1\}$</code> 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.</p>