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Concerning the proof that $\zeta(3)$ is irrational, Van der Poorten famously noted that

"Apéry's incredible proof appears to be a mixture of miracles and mysteries".

Indeed, many ideas introduced in Apéry's proof such as $\zeta(3)=\frac{5}{2} \sum_{k=1}^{\infty}{\frac{ (-1)^{k-1}} {\binom {2k}{k}k^{3}}}$ and the recurrence $n^3u_n + (n-1)^3 u_{n-2} = (34n^3-51n^2+27n-5)u_{n-1}, n\geq2$ amazed contemporary mathematicians (although the fast-converging series was already derived several years earlier by Hjortnaes).

What are other examples of "miraculous" proofs, whose ingredients amazed mathematicians of their time? In particular, proofs such that:

  • a large portion of the Theorems involved in the proof represent entirely new ideas,
  • these Theorems are applicable to a wide area of mathematics,
  • the general atmosphere among contemporary mathematicians was a mixture of surprise and awe ("where did this come from?").
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closed as primarily opinion-based by YCor, Igor Pak, Gro-Tsen, Mark Sapir, Suvrit Jul 15 '18 at 11:57

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ While I would certainly enjoy reading the answers to this question, I do think it is far too broad to be appropriate and have voted to close. $\endgroup$ – Noah Schweber Jul 13 '18 at 14:12
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    $\begingroup$ It would be fun to reopen if the scope could be refined a bit. $\endgroup$ – xuq01 Jul 13 '18 at 14:16
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    $\begingroup$ I don't think it is too broad. There many big list questions like that already on MO. $\endgroup$ – Abdelmalek Abdesselam Jul 13 '18 at 14:18
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    $\begingroup$ I have a mixed feelings about the question. I still think it is too broad, but on the other hand one may expect many interesting answers so after the question was edited, I voted to repoen it. $\endgroup$ – Piotr Hajlasz Jul 13 '18 at 15:28
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    $\begingroup$ What's more important, strict adherence to arbitrary rules, or applying a little leeway and doing something that apparently everyone will enjoy? $\endgroup$ – Najib Idrissi Jul 13 '18 at 16:32

13 Answers 13

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The best one I can think of is Paul Cohen's proof of the independence of the continuum hypothesis. That certainly had a huge impact, and I may be wrong, but my impression is that set theorists at the time felt that the proof came out of nowhere. In retrospect maybe one can find some hints of the technique in earlier work, but again, at the time I think it was seen as totally new.

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    $\begingroup$ +1 and/but RE: "hints of the technique in earlier work" I think the priority method can be seen as such an example: cf. e.g. Kreisel's letter to Gödel in 1963 mentioned at the start of MO 124011. $\endgroup$ – Benjamin Dickman Jul 14 '18 at 1:08
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    $\begingroup$ @BenjaminDickman And on that note I think the Friedberg-Muchnik theorem itself would also constitute a good answer to this question; the idea of injury, and of managing injury through priority, was totally new and completely changed the face of the subject. $\endgroup$ – Noah Schweber Jul 22 '18 at 2:52
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Thurston's geometrization conjecture came out of the blue around 1980. He proved it for Haken manifolds in 1982. In his famous essay, On proof and progress in mathematics, he says:

"It was a hard theorem, and I spent a tremendous amount of effort thinking about it. ... Neither the geometrization conjecture nor its proof for Haken manifolds was in the path of any group of mathematicians at the time—it went against the trends in topology for the preceding 30 years, and it took people by surprise."

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I nominate Jack Silver's proof that, if the generalized continuum hypothesis is false, then the first counterexample cannot be a singular cardinal of uncountable cofinality. In the first place, the result was, as far as I know, totally unexpected. The general feeling was that Easton's results on violations of GCH at regular cardinals should extend to singular cardinals. Secondly, the proof was an unexpected mixture of combinatorial set theory and nonstandard models. (Later, other proofs were found that avoid the nonstandard models.)

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I think Donaldson's work on gauge-theoretic invariants for smooth 4-manifolds was also considered miraculous by topologists at first (as probably was Witten's introduction of SW invariants which are roughly equivalent in power to Donaldson invariants and MUCH easier to work with).

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    $\begingroup$ I agree that Donaldson’s theorem Cameron out of nowhere. I don’t think anyone thought Yang-Mills could be used to prove topological theorems. $\endgroup$ – Deane Yang Jul 14 '18 at 3:12
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I quote from Michael Monastyrsky's book Modern Mathematics in the Light of the Fields Medals.

The most important of Margulis' results is his proof of Selberg's conjecture that a certain class of discrete groups is arithmetic. … Margulis' papers display his exceptional originality. By invoking ideas from widely separated branches of mathematics, he proceeds to the goal by the simplest possible route. One of these theorems, which was central to proving the theorem on arithmetic subgroups, was announced several years before the full proof appeared. Despite efforts of many leading experts in this area, the result could not be reproduced independently. The nature of Margulis' original proof was very simple and, to the experts, unexpected.

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It's not really a theorem and maybe not miraculous, but I think the way Gromov, in his book Partial Differential Relations solves an underdetermined system of PDEs (fewer equations than unknown functions) is hilarious. Normally, a differential equation is solved using integration in one way or another. Gromov shows that many sufficiently underdetermined systems can be solved explicitly in terms of the coefficients and their derivatives. The simplest example is to solve for $u$ and $v$ satisfying

$$ \partial_x u + v = f $$

The simplest nontrivial example, just an ODE, is:

$$ au' + bv' = f $$

Note that if $a$ and $b$ are constant, then the only way to solve this is by integration. However, if $a$ and $b$ are functions satisfy a nondegeneracy condition, then a solution $(u,v)$ can be written explicitly in terms of $a$, $b$, $f$, and their derivatives. See my note on this.

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    $\begingroup$ See likewise J. L. Synge, solving the Kepler problem: “To integrate, we first differentiate, obtaining...” $\endgroup$ – Francois Ziegler Jul 14 '18 at 14:18
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    $\begingroup$ +1. Gromov's PDR seems to hide many surprising gems in there. The very premise of h-principles seem absurd, hence exciting, to me, which is that one can approximate (but that approximation is usually not of high regularity) non-integrable sections (of certain jet bundles) by integrable ones (at the cost of $\epsilon$-modifying/corrugating the domain of the sections). But surely that's because I am an amateur and not used to these circle of ideas. One of the proofs happens to simply be an interpolation trick... $\endgroup$ – Balarka Sen Jul 15 '18 at 7:39
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My favorite example is Gunther’s radically simpler proof of the Nash isometric embedding theorem, which came out of nowhere.

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    $\begingroup$ Can you provide a reference for Gunther's result? $\endgroup$ – Piotr Hajlasz Jul 14 '18 at 1:06
  • $\begingroup$ The original paper is: MR1029846 Reviewed Günther, Matthias On the perturbation problem associated to isometric embeddings of Riemannian manifolds. Ann. Global Anal. Geom. 7 (1989), no. 1, 69–77. (Reviewer: S. B. Klimentov) 58C15 (53C42) $\endgroup$ – Deane Yang Jul 14 '18 at 13:33
  • $\begingroup$ He also spoke on it at the ICM: MR1159298 Reviewed Günther, Matthias Isometric embeddings of Riemannian manifolds. Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), 1137–1143, Math. Soc. Japan, Tokyo, 1991. (Reviewer: Chong Kyu Han) 53C42 (58G30) $\endgroup$ – Deane Yang Jul 14 '18 at 13:34
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    $\begingroup$ Terry Tao wrote about on his blog: terrytao.wordpress.com/2016/05/11/… $\endgroup$ – Deane Yang Jul 14 '18 at 13:36
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I think Stanley's introduction of tools from commutative algebra to study convex polytopes in his proof of the upper bound theorem for simplicial spheres was completely novel and miraculous; for a historical account see: http://www-math.mit.edu/~rstan/papers/ubc.pdf.

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Gromov's theorem on groups with polynomial growth has, I think, an amazing proof. It's where he introduced Gromov-Hausdorff distance and showed, using the solution to Hilbert's fifth problem, that a sequence of discrete groups, under the right assumptions, converges to a Lie group.

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I am surprised that nobody has mentioned the Riemann hypothesis and its consequences. I can never hide my fascination that somebody could manage to prove that the distribution of prime numbers and many other problems in number theory, all boil down to proving that the roots of a seemingly nice function in the complex plane have equal real parts.

I wish I could some day travel back in time and ask him really, what kind of sorcery is this?

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I've always been impressed by Lovasz's cancellation theorem among finite relational structures. Some details are at https://mathoverflow.net/a/269610 and nearby.

Gerhard "Search MathOverflow For More Answers" Paseman, 2018.07.13.

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Nash equilibrium. I don't think before that anyone guessed Analysis would have an application in Game Theory.

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    $\begingroup$ Not sure if this answer fits the "miraculous" category -- at least if you follow the history behind it (and the story of what von Neumann had to say etc.) $\endgroup$ – Suvrit Jul 13 '18 at 21:35
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Going further back, Galois correspondence came out of nowhere.

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    $\begingroup$ This seems wrong to me. Galois's contemporaries did not react with amazement and awe but with confusion and requests for further explanation. Consider Edwards's sympathetic presentation (ams.org/notices/201207/rtx120700912p.pdf), and look at his corrected Proposition 2: it's far from what we would call a Galois correspondence in an undergraduate class, and rather than coming out of nowhere, it looks like the tradition of studying permutations in solving equations. $\endgroup$ – Matt F. Jul 14 '18 at 3:31

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