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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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I have flagged this question to be made CW. – JavaMan Jan 15 at 2:55
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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. – Giuseppe Negro Jan 15 at 3:17
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@GiuseppeNegro If it really happens, there is nothing to be dispelled. I think it is 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. – Alexander Gruber♦ Jan 15 at 3:37
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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. – dwarandae Jan 15 at 4:02
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@Jonathan, independently of what? You have to reconcile that view with Newton's remark that he stood "on the shoulder of giants." – alancalvitti Jan 15 at 4:04
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Marjorie Rice surprised Martin Gardner and most 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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Can you point to someplace that describes these proofs and why they were wrong? The wikipedia page doesn't seem to mention them. – HappyEngineer Jan 30 at 20:38
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Goedel'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 Goedel. 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 Goedel did not build on pervious 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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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. – Eric Stucky Jan 15 at 2:56
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. – Ittay Weiss Jan 15 at 2:58
Alright, that's fair. – Eric Stucky Jan 15 at 3:04
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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.

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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. – Alexander Gruber♦ Jan 15 at 2:43
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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 – B.D Jan 15 at 2:48
I like that. +1 – Alexander Gruber♦ Jan 15 at 2:58
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/… – B.D Mar 20 at 10:54
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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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RE: Cohen's forcing. See the comments in my answer. – B.D Jan 15 at 19:51
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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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