Results that came out of nowhere.

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.

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.

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.

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.