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Some people ask questions here out of simple curiosity. But some ask them because they are working on a research project, come up with a question they need to know the answer to, and think that the answer is probably known. In the past, one had to search for the right person to tell you the answer or trawl through lots of books and articles, often not knowing quite where to look. Mathoverflow ought to be a far more efficient way of doing it.

So my question here (asked out of simple curiosity) is whether there are some good examples of people using mathoverflow in an essential way to solve a research problem. The best example would be a story such as this: you struggled to solve a problem, you identified a statement that you thought would be helpful, you asked about it on mathoverflow, you got an answer, and the answer was just what you needed to complete your research project.

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I think this is more appropriate for meta. Mathoverflow is for math, not for posting questions about mathoverflow. The rules may have changed, I don't know, but I'm just letting you know. –  Harry Gindi Jan 15 '10 at 12:04
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This question is both about math and about MO unlike a typical meta item. You can compare it to the question: "Have blogs led to mathematical breaktroughs" which is unsuitable to meta, but is suitable as a soft mathoverflow question. (I prefer questions in mathematics over questions about mathematics' infrastructure but the later can be useful too.) Regarding the question itself, we can compare MO to email or Tex. WISELY used it can free time for whatever leads to breakthroughs. Finally, I prefer the term "advances" or "substantial advances" rather than "breakthroughs". –  Gil Kalai Jan 15 '10 at 13:55
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Although I agree that this is probably better for the Meta (there have been related discussions there), I do think this is a great, question. –  Theo Johnson-Freyd Jan 15 '10 at 17:02
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Oh, my intuition is that this is the type of question that, if it is going to be here rather than on the Meta, should be community wiki. But I'm not really sure what the aesthetic is for that. –  Theo Johnson-Freyd Jan 15 '10 at 17:04
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All users of MathOverflow read MathOverflow; a small minority reads meta as well. Any question of interest to a wide class of mathematicians (such as this one) has a right to be posted here. –  Tom LaGatta Jan 15 '10 at 22:11

11 Answers 11

I have an example almost of the sort you are looking for, but not quite with the happiest ending (at least for me!). I was struggling to solve a problem. I identified a statement that would be helpful. I asked on mathoverflow. I got an answer. It turns out that both the statement and my intended application were about to appear in an article by some other authors.

In this instance mathoverflow has saved me a great deal of effort simply by allowing me to not work on the problem and read instead someone else's solution. Without mathoverflow it is very unlikely that I would have heard of the article before it was published. (It doesn't seem to be on the arxiv.)

In general, it seems to me that mathoverflow is an extremely efficient way of getting expert advice on research-level questions.

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Hey Tim. I'm not so sure whether my model of how "proving a theorem" works is the same as yours. But let me give some kind of an example of something and you can take it or leave it. I'm writing a paper with Toby Gee at the minute, and we're both number theorists, and the arguments in the paper are "robust" but the details need checking. We're now at the point where we're writing up technical calculations and these technical calculations are mostly in the area of representation theory of reductive algebraic groups, an area which I think it's fair to say that neither Toby nor I would call ourselves experts in. So we have this overall "robust" argument, and a write-up that exists but occasionally says "lemma: (statement in representation theory); proof: TO BE ADDED". We 'know' these lemmas are true because they fit into our overall picture, but occasionally when I write one of these things up I can't go from my intuition to a rigorous proof. Here's an example of an occasion when I got stuck:

This question of mine.

Ben Webster made a crucial remark that enabled me to finish the argument, so that lemma went from "must be true but proof not yet written or even discovered by authors" to "lemma proved".

So if I were interpreting your question in a particularly anal way, one might argue that had this lemma been the last of the lemmas we need to write up the proof of, then Ben's contribution might be "just what I needed to complete my research project". Unfortuately there are several more to go :-).

Having said all that, it's not clear to me that MO was "crucial" to solving the problem. I could have worked more on the problem until I'd done it myself. I could have asked one of the representation theorists in my own department. I could have left it and hoped that my co-author sorted it out. All of these would have been viable approaches. Why did I ask at MO? Simply because I am sick of writing this paper and asking at MO was by no means the only way of solving the problem, but I had high hopes that it would be the quickest.

Kevin

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My own feeling about use of the internet is that one can more or less prove that it isn't "crucial" in the fully anal sense, because there are other means of communication. But it can be crucial in the weaker sense of being so much more efficient that it makes the difference between progress not happening and happening. Your example isn't quite like that, since you'd presumably have completed your research in reasonable time without mathoverflow, but it's quite far along a spectrum towards that. –  gowers Jan 15 '10 at 12:18
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A lot depends on the researcher's environment. I am not physically located near the university I'm working on a physics dissertation for (and even when I was, there was only one person there who could reliably answer even a few questions I would have about mathematics), since I moved once completing quals. I only know a couple of folks at the local research universities, and none who have expertise in what I'm working on now (though there are some around). So MO is a great place for me to make progress when I get stuck on something, especially trivial sorts of things. And that does happen. –  Steve Huntsman Jan 15 '10 at 15:55

My question about graph norms being cyclotomic received two inconclusive answers, but also resulted in another mathoverflow user contacting me and Noah Snyder about the problem. There's now an almost finished paper that completely answers the question.

I'm not sure if this exactly fits the mold of your question --- it's not that mathoverflow helped with a lemma, but rather that mathoverflow brought a difficult question to the attention of the right person.

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This MO question Walsh Fourier Transform of the Möbius function was aimed toward a certain conjecture (which was offered a few days earlier on my blog and Dick Lipton's blog) extending the prime number theorem (the constant one function is asymptotically orthogonal to the Mobius function) to certain class of low-complexity functions. (Relpacing the constant one function.)

A few days after being posted Ben Green indicated a solution to the conjecture and a few days later the complete details appeared in a lovely short paper that Ben wrote. (Ben also indicated a method that may lead to a complete solution of the MO question, but this may require more work.) This is a substantial advance.

In this case, this was not a question Dick and I struggled with but rather a question we hoped to attract an expert who may find it interesting.

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John Pardon has solved the Hilbert-Smith conjecture for group actions on 3-manifolds. Lemma 2.17 of the paper was based on the answer to this mathoverflow question. I was quite surprised to receive an e-mail a few months after answering the question with a preprint resolving the conjecture, especially since I did not know the identity of the MO user or for what purpose the question was intended.

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See this very similar discussion on meta.MO.

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The question of mine mentioned on that meta thread is a perfect fit to what gowers is asking, I think. I had an open question which was important in my research; I asked it on MO; Mark Meckes solved it; putting his solution together with some other stuff closed the open question; now this line of research can progress. –  Tom Leinster Jan 15 '10 at 12:50

My feeling is there have been a number of observations that could be expanded into short papers, such as Kevin's example of a smooth proper scheme over Z with no section. I expect though that most of the time, people won't bother.

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The answer to this question has saved me a lot of time and effort on my current research. Besides that, it helped me get a better understanding of the structures involved.

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PS. Would someone please consider upvoting the answer? It was really helpful to me. –  Steve Huntsman Jan 15 '10 at 15:09
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BTW just to be 100% clear: I don't mean my answer here (or there). –  Steve Huntsman Jan 15 '10 at 22:18

Chris's answer to my first question definitely cleaned up some statements in my current project.

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There was a question which could be interpreted as a question about Latin squares. I presented at a problem session at our research group meeting at Monash, where it was completely resolved (at least, the part I'm directly interested in).

PS: Although, this is hardly a breakthrough.

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David Speyer's answer to my question has helped me to finish my Senior Project paper. But it is more at the level of graduate level homework exercise or student research than serious breakthrough. In case anyone wishes to look at it, I proved David's basis really works as well as provided computational proof for some other small rank examples.

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