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Most big results in mathematics are built on years and years of groundwork by the author and other mathematicians, such as Wiles' proof of FLT or the classification of finite simple groups. Nevertheless, there is this big romantic idea in pop culture where a young genius comes up with a brilliant proof of a major conjecture, based on some creative new idea which flies in the face of the mathematical community.

What are some examples where this has actually happened? That is, which results stem from independent work by a mathematician who came along "out of nowhere" and solved a huge problem by surprise through nonstandard techniques?

I read that the recent proposed proof of the ABC conjecture comes from years of autonomous theory by Prof Mochizuki, most of which ventures far outside of the current literature; this would seem to qualify as one example provided the proof turns out to be true.

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    $\begingroup$ A totally personal remark: I don't like very much this kind of questions. That "romantic idea in pop culture" should be dispelled, not encouraged. $\endgroup$
    – Giuseppe Negro
    Commented Jan 15, 2013 at 3:17
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    $\begingroup$ @GiuseppeNegro If it really happens, there's nothing to be dispelled. I think it's neat to hear about sudden historic advancements. But I do agree that the role of non-celebrity mathematicians should receive more attention in the media. $\endgroup$
    – Alexander Gruber ♦
    Commented Jan 15, 2013 at 3:37
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    $\begingroup$ The ABC conjecture has not been proved, Vesselin Dimitrov and Akshay Venkatesh found an error in the proof in October, 2012 and Mochizuki accepted the mistake. $\endgroup$
    – dwarandae
    Commented Jan 15, 2013 at 4:02
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    $\begingroup$ I'd prefer not to limit the question to young geniuses. What about all of us aged geniuses? $\endgroup$
    – Andreas Blass
    Commented May 24, 2013 at 18:37
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    $\begingroup$ @dwarandae according to Wikipedia, Mochizuki posted a corrected version of his set of papers in March 2013. So it's still up in the air? $\endgroup$
    – Stahl
    Commented May 24, 2013 at 21:43

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Gödel's completeness and incompleteness theorems I believe qualify as results coming out of nowhere. Hilbert declared as one of the most important tasks for the 20th century a foundations for mathematics proved impossible by Gödel. So, this is not a case where a brilliant mathematician comes with a proof of a major conjecture but instead crumbles a major conjecture. As far as I know Gödel did not build on previous work for his incompleteness theorems.

Grothendieck's work in algebraic geometry transformed the entire field and as far as I understand this was completely his doings.

Galois' work should perhaps be number one on the list. Certainly, nobody saw him coming and again as far as I know he developed almost all of his results on his own.

Then there is Ramanujan who proved in complete isolation an unbelievable range of results in number theory. He claimed that these results were given to him in his sleep by a goddess, so I guess that came out of nowhere.

Hamilton's creation of the quaternions can count as coming out of nowhere, at least considering the rudimentary development of abstract algebra at his time.

The discoveries of projective and hyperbolic models for planes that finally settled the quest for a proof of Euclid's proof might count as coming out of nowhere as no previous models were in existence (disregarding the fact that we all walk on a pretty good approximation of a sphere).

Cantor's creation of the heaven of set theory certainly fits the requirements (in particular, the uncountability of the real numbers --> indirect proof of existence of non-algebraic numbers).

The irrationality of the square root of 2 is a well known historical event but it is a bit hard to discren who precisely proved it so perhaps it did build on previous work, I do not know.

Since you mentioned popular movies depicting such acts of heroic mathematics, the development of game theory by Nash I believe fits the bill.

I'm sure there are more examples, but I'll stop here.

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    $\begingroup$ The one I have to take objection to is Hamilton. A lot of people were thinking about numbers that encode three-dimensional rotations, and he just seems to have realized that the solution was in four dimensions before the next guy did. $\endgroup$
    – Eric Stucky
    Commented Jan 15, 2013 at 2:56
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    $\begingroup$ yes, I agree Eric. The reason I included it in the list is because of the famous story according to which the idea came out of nowhere to him, forcing him to stop along the way and write it on a bridge somewhere. $\endgroup$
    – Ittay Weiss
    Commented Jan 15, 2013 at 2:58
  • $\begingroup$ Alright, that's fair. $\endgroup$
    – Eric Stucky
    Commented Jan 15, 2013 at 3:04
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    $\begingroup$ But he had been thinking about the problem for ten or fifteen years before the bridge episode, so there is only a very limited sense in which it "came out of nowhere". $\endgroup$
    – MJD
    Commented May 24, 2013 at 18:53
  • $\begingroup$ I don't think the discovery of non-Euclidean geometry counts as coming out of nowhere - didn't Riemann propose the new models when he was already pretty well established? Plus, people were already doing hyperbolic geometry, just without realizing it (see: Giovanni Saccheri). $\endgroup$
    – MartianInvader
    Commented May 24, 2013 at 21:20
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Marjorie Rice surprised Martin Gardner and many of the readers of Mathematical Games when she found new pentagon tilings in 1977.

With no formal training in mathematics beyond high school, she (Marjorie Rice) uncovered a tenth type of pentagon.... Her method of search was completely methodical, beginning with an analysis of what was already known.1

Several papers had supposedly proven that such tilings were not possible. Rice found three additional pentagon tilings in the years that followed.

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    $\begingroup$ Can you point to someplace that describes these proofs and why they were wrong? The wikipedia page doesn't seem to mention them. $\endgroup$
    – HappyEngineer
    Commented Jan 30, 2013 at 20:38
  • $\begingroup$ Seconded - my strong impression from having looked at the material previously is that everyone 'knew' that was all the pentagons there were, but that no one had bothered to prove that the list was ever complete. (And I'm not sure I would argue that Rice 'solved' anything, so much as that she discovered additional examples - but that's perhaps just a matter of semantics.) $\endgroup$
    – Steven Stadnicki
    Commented May 24, 2013 at 20:13
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    $\begingroup$ Update: All pentagon tilings have now been found :) $\endgroup$
    – Akiva Weinberger
    Commented Dec 13, 2019 at 5:38
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I remember when Apéry's proof that $\zeta(3)$ is irrational appeared. There had been essentially no progress on values of $$1+\frac{1}{2^n}+\frac{1}{3^n}+\frac{1}{4^n}+\cdots$$ for odd $n$ since the time of Euler.

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Cantor's diagonal argument ${}{}$

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Shelah's proof that the Whitehead problem is independent of ZFC.

And while we're at it to some extent Cohen's technique of forcing. People already knew that the axiom of choice could be negated using atoms, and people knew how to have relative consistency and unprovability results, but Cohen's forcing was special in that it didn't generate some pathological model (like compactness arguments would) but rather take a nice model of SFC and produce another nice model of ZFC.

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    $\begingroup$ RE: Cohen's forcing. See the comments in my answer. $\endgroup$
    – Benjamin Dickman
    Commented Jan 15, 2013 at 19:51
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Probably Ramanujan's work would be the easiest answer to come up with for your question.

However, I take slight issue with your classification of Mochizuki's work as being an example of "coming out of nowhere." It may be true that he worked for a while mostly on his own, but his results certainly did not come from nowhere. See this MO post and its various answers for more about the philosophy behind his work and the results that preceded it.

Edit: Okay, yeah, if Mochizuki's work is ever classified, then it probably ought to fall under a reasonable definition of "coming out of nowhere."

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    $\begingroup$ I have read that post, actually. I was under the impression that the reason Mochizuki's work was still being verified is that it builds on a wealth of theory he developed essentially by himself for the last 20+ years. I don't mean to say that it does not build off of other results also. $\endgroup$
    – Alexander Gruber ♦
    Commented Jan 15, 2013 at 2:43
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    $\begingroup$ Yes, I recognize your name and suspected you had. I suppose it depends on how you conceive of "coming out of nowhere." When I read your question, I wondered whether Cohen's technique of "forcing" might be a good answer. In fact, he wrote a paper on his own discovery of forcing, so I will leave a link and let you decide for yourself: projecteuclid.org/euclid.rmjm/1181070010 $\endgroup$
    – Benjamin Dickman
    Commented Jan 15, 2013 at 2:48
  • $\begingroup$ I am increasingly convinced that Cohen's forcing is not a result that came out of nowhere. See, for example, this MO post: mathoverflow.net/questions/124011/… $\endgroup$
    – Benjamin Dickman
    Commented Mar 20, 2013 at 10:54
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Shannon's 'Mathematical Theory of Communication' was a work that came out of 'nowhere' in that he defined a paradigm and then modeled it perfectly and completely... (IMO)

There were others interested in the field but no one had encompassed it or envisaged it correctly.

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I'm not sure this is correct, but I can't think of any precursors to Stephen Cook's invention of NP-completeness. With a single paper in 1971 he invented the idea of NP-completeness, previously unimagined, and showed that the satisfiability problem was NP-complete; this revolution has dominated the study of algorithms ever since. The same paper posed the $P=NP$ question that has remained one of the foremost questions of computer science.

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    $\begingroup$ I should add, however, that Leonid Levin independently discovered NP-completeness around the same time, so perhaps it was less of a surprise at the time than I think it was. $\endgroup$
    – MJD
    Commented May 24, 2013 at 19:01
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Smale's proof of sphere eversion and the proof of the Poincaré conjecture for dimensions 5 and higher were unexpected and completely out of the blue.

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Perhaps the recent proof that there are infinitely many primes with a (fixed) bounded gap qualifies? (Although this story is not necessarily a happy ending for all, it has still caused quite the buzz!)

For those interested, I may as well add the actual work (subscription/campus IP/proxy required)

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  • $\begingroup$ I don't understand why you say it is not necessarily a happy ending for all. I probably read the article too fast. +1, I had missed this story until today. $\endgroup$
    – Julien
    Commented May 24, 2013 at 21:57
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    $\begingroup$ @julien, the second link takes some time to load fully and, eventually, goes to a comment by a reader. It's not (necessarily?) relevant to the math side... but I'll post it here as well: "So sad another promising sandwich artist lost to the lure of Mathematics. Oh well maybe he wasn't cut out for it. He should not be disappointed with himself." $\endgroup$
    – Alex
    Commented May 24, 2013 at 23:03
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Vapnik and Chervonenkis work on classification and learning via Support Vector Machines (SMV) is said to have come "from left field."

Though SMVs are more technology than math - since the kernels have to be engineered to specific applications (and further SVMs are but one method out of a long list applied to learning), their definition VC dimension is now a basic part combinatorial geometry.

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