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In mathematics, it is common that theorems/results and problems appearing dull in one generation get revitalized and become the center of research in another one.

Sometimes conjectures that are thought to be untouchable become resolved within years.

I would like to know if there were conjectures that did not appeal to a large number of mathematicians but when they were resolved( maybe partially), the techniques used or the result itself touched upon many areas. On the other hand, are there also conjectures which were expected to have dramatic impact but upon resolution did not meet to that expectation?

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At the very least, this question ought to be community wiki. I think that the bounty is completely inappropriate. Indeed, I would consider closing this as inappropriate for this site (being somewhat of an open-ended discussion question) except that I can't do so due to the bounty. I've started a thread on meta: tea.mathoverflow.net/discussion/427/… –  Loop Space Jun 7 '10 at 12:10
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Of course you realize this means war. –  Bugs Bunny Jun 8 '10 at 16:53
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I do not reckon it inappropriate. Evidently, I have seen posts/comments of reputed mathematicians on questions like this one. I hope there is a lot to be learned from the answers. Maybe, one may change one's views or reconsider one's thoughts about mathematical problems in relation to their purported consequence from such historical experiences of others. –  Unknown Jun 10 '10 at 6:32
    
Good answers, I can't choose among them. –  Unknown Jun 14 '10 at 6:19
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The more experienced users of MO could foresee that your question would generate a big list of good answers, which would be difficult to choose among. That is why they suggested that it be made community wiki and the bounty removed. Hopefully, with this experience, you will in the future be more receptive to suggestions from more experienced users. –  Timothy Chow Jun 15 '10 at 1:56

4 Answers 4

up vote 13 down vote accepted

An example of an important solution to a little-known problem might be Frank P. Ramsey's "On a problem of formal logic" in Proc. London Math. Soc. 30 (1930) 264-286. The problem was in logic and not well-known even to logicians, but Ramsey's solution was taken up by combinatorialists (notably Erdős and Szekeres) and it grew into the important field now known as Ramsey theory.

{Added later] An example of the contrary type is Hilbert's fifth problem. This was a well known and difficult problem, worked on by eminent mathematicians such as von Neumann and Pontryagin, and it took more than 50 years to solve. Yet, by the time it was solved it seemed to be no longer in the mainstream of Lie theory, and books on Lie theory today make little mention of it.

PS. I agree that this question should be community wiki.

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Hi John, Decidability problems were central to logic for a big part of last century. Why do you say that Ramsey's decidability result was on a problem that was not well-known? (I am not questioning your claim. I'm genuinely curious, so as not to claim something false when teaching Ramsey's theorem, for example.) –  Andres Caicedo Jun 7 '10 at 23:20
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Andres, I may be wrong, but I think there was very little reaction to Ramsey's theorems from logicians at the time. (Maybe they were too busy absorbing Gödel's theorems.) The first people to take an interest were Erdős and Szekeres in their 1935 paper in Compositio Math., "A combinatorial theorem in geometry". Among other things, Erdős and Szekeres gave a much simpler proof of the finite Ramsey theorem, and they noticed the connection between infinite and finite Ramsey theorems via the Kőnig infinity lemma. –  John Stillwell Jun 8 '10 at 0:08
    
Point of clarification: The "problem of formal logic" referred to in the title of Ramsey's paper was the decision problem for first-order logic. This problem was certainly famous (and eventually resolved negatively by Church). However, the most lasting contribution of Ramsey's paper was not his partial result on the decision problem, but a result he proved along the way that we now know as "Ramsey's theorem." Probably because of these circumstances, Ramsey's theorem was largely overlooked by combinatorialists for many years, even though Ramsey flagged it as having "independent interest." –  Timothy Chow Jun 15 '10 at 1:50
    
Thanks, Tim. I should have made it clear that Ramsey's "solution" was only partial (and it could not have been complete because, as you say, the problem is unsolvable). Perhaps logicians were not impressed because they were still hoping for a complete solution. –  John Stillwell Jun 15 '10 at 2:46
    
I should point out that the work on Hilbert's fifth problem was a major component in Gromov's proof of his famous theorem on groups of polynomial growth, and more recently was used by Hrushovski to obtain some deep results on finite sets of bounded doubling. My experience has been that any really deep piece of mathematics tends to find its uses eventually... –  Terry Tao Aug 20 '10 at 1:30

An open problem whose solution did not deliver what many people were hoping for was the elementary proof of the prime number theorem. Although this was a fantastic achievement by Erdos and Selberg, it has not led to further dramatic breakthroughs in number theory, at least not to the extent that many people were hoping.

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Kummer, in the 1850s, proved the p-th power reciprocity laws for regular primes. He conjectured that his formulation was valid for arbitrary primes, but the only one who mentioned this conjecture and hinted at a way of solving it was his student and friend Kronecker. Only when Hilbert, at the end of the 1890s, proved a quadratic reciprocity law in number fields with even class number did it become clear what to do, and within the next 15 years Furtwängler followed Hilbert's path and proved the full p-th power reciprocity law. These results didn't appeal to many mathematicians, and only when Artin found a way of formulating the reciprocity law in a conceptual way did it become a widely known and widely used result.

Certainly one reason why Kummer's conjecture was largely neglected was that the techniques for solving the problem were lacking. In addition, a lot of people probably did not think it was interesting because the problem could not be put into some conceptual framework since Frobenius published his results on the Frobenius automorphism at about the same time Hilbert was working on these problems.

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One can contrast Hilbert's 7th problem (http://en.wikipedia.org/wiki/Hilbert%27s_seventh_problem) on transcendence with his views on Fermat's last theorem. These are reported somewhere, namely that the 7th problem would be harder to solve, since FLT would probably be solved soon. Now the 7th problem was solved (or at least the better-known part?), and it would be unfair to call the Gelfond–Schneider theorem shallow, but it was only one advance in transcendence theory which required new ideas after that. On the other hand we now know with hindsight that the "modularity theorem" approach to FLT goes really deep; this wasn't something that wasn't at all apparent until about 25 years ago. There might have been a "cheap" proof of FLT that showed that the existing criteria on p did rule out all cases: but something very different happened.

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How come only one advance in transcendence theory after Gelfond-Schneider resulted in two Fields medals (Roth and Baker)? Or did you mean "one advance from Hilbert to G-S"? –  Victor Protsak Jun 5 '10 at 9:33
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Roth's result extended the work of Thue and Siegel in diophantine approximation. It was Baker's work that extended what Gelfond did (from two logarithms of algebraic numbers not commensurable to lower bounds for any linear form in logarithms). I think the point here is that auxiliary functions in several complex variables were needed, and new ideas. –  Charles Matthews Jun 5 '10 at 10:59
    
I believe we cannot judge the value of 7th problem properly until the Schanuel's conjecture is resolved. –  Alex Gavrilov May 9 '11 at 13:08

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