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By a blackbox theorem I mean a theorem that is often applied but whose proof is understood in detail by relatively few of those who use it. A prototypical example is the Classification of Finite Simple Groups (assuming the proof is complete). I think very few people really know the nuts and bolts of the proof but it is widely applied in many areas of mathematics. I would prefer not to include as a blackbox theorem exotic counterexamples because they are not usually applied in the same sense as the Classification of Finite Simple Groups.

I am curious to compile a list of such blackbox theorems with the usual CW rules of one example per answer.

Obviously this is not connected to my research directly so I can understand if this gets closed.

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The simpler proofs are still at least 20 pages of fairly technical mathematics though. –  Karl Schwede Jun 13 '12 at 21:38
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Domain of use is important here. Many theorems are invoked by physicists who have no idea of the actual proofs. –  Steve Huntsman Jun 13 '12 at 22:05
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@Zsbán: That theorem has nice consequences, e.g., in finite geometry. But treating it as a blackbox is just laziness, since the proof is just a couple of pages of basic graduate algebra. –  Felipe Voloch Jun 15 '12 at 0:19
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The classification is used by people working on permutation groups and graph theory all the time. Also they are used in profinite group theory. –  Benjamin Steinberg Jun 16 '12 at 19:21
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In my mind I was hoping for things used in at least 100 papers and understood in all technical detail by fewer than 5% of people in the general area to which the theorem belongs. But it need not be this rigid. –  Benjamin Steinberg Jun 17 '12 at 3:08
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closed as no longer relevant by Benjamin Steinberg, Bill Johnson, Felipe Voloch, Will Jagy, Asaf Karagila Sep 15 '12 at 9:38

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60 Answers

Perhaps the existence of "Tarski Monsters" qualifies as a "blackbox" theorem. The theorem is that for sufficiently large prime numbers $p$, there exist infinite groups $G$ such that every proper nonidentity subgroup has finite order $p$. Such a Tarski monster is clearly 2-generated and has finite exponent, so it provides counterexamples to the Burnside problem. Also it is a simple group of prime exponent and it provides counterexamples for many other attempts to generalize properties of finite groups to groups in general.

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Cech and Sheaf (derived-functor) cohomologies are isomorphic on a paracompact space $X$ with the sheaf being, for example, $\underline{\mathbb{C}}^*_M$, the sheaf of $\mathbb{C}^*$-valued functions on $X$.

The proof uses partititions of unity along with hypercohomology and results from spectral sequences. If this ISN'T a black box theorem, I'd love a concise explanation :)

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Doesn't Zorn's Lemma count? Of course in ZF this is not a Theorem (rather it is undecidable), but in ZF+AC it is a real Theorem which is often mentioned without proof, especially in classes outside of mathematical logic. For example, in commutative algebra it is quoted in order to get enough maximal ideals in rings, etc.

Of course it is not hard to understand the proof of AC => Zorn, but many students take this on faith. I don't know if this also applies to mathematicians.

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Okay, but in ZF+Zorn's lemma it's a tautology! I don't think in practice most people use specifically the fact that AC implies Zorn's lemma but just the fact that it's generally considered okay to prove results that depend on Zorn's lemma. –  Qiaochu Yuan Jun 14 '12 at 10:30
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I've already posted this as a question on MathOverflow, but it appears that everyone working on the Ricci flow, as well as other geometric heat flows, takes it for granted the existence in short time of a solution to a nonlinear parabolic PDE on a vector bundle over a complete Riemannian manifold. I have not been able to find a complete proof for even a linear parabolic PDE.

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In theoretical computer science, possibly the best example is the PCP Theorem: it's used all over the place, from cryptography to quantum computing, yet very few of us understand the details (especially for the strong, "modern" versions of it).

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Her proof is certainly well-understood at a high level, and it's also certainly better-understood than the algebraic proof. But by the standard of how many people could fully reconstruct her proof within (say) two months, if locked in a room by a mad scientist with only pencils and blank paper ... well, the requisite experiment hasn't been done, and I shouldn't speak for everyone else, but I wouldn't want to be put to the test. :-) –  Scott Aaronson Jun 19 '12 at 6:15
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Deligne's construction of Galois representations corresponding to modular forms.

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A perfect example! –  Joël Jun 28 '12 at 12:52
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Most mathematicians can recite the construction of a Vitali set and state that the axiom of choice is needed. Very few of them would know to describe the proof that the axiom of choice is really needed, i.e. Solovay's model (or even the Feferman-Levy in which every set is Borel).

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There are a lot of complicated 'motivic' statements. I would say that the Milnor Conjecture (and its generalization, the Bloch-Kato conjecture) are easy to understand and apply; yet its proof is very hard.

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The Lovasz Local Lemma gives a very simple criterion for when certain random events have positive probability. Almost all applications of the Lemma automatically have an algorithm to find such events. The details of these proofs (especially in their most general forms) can be messy, but the LLL criterion is so simple you can use it basically as a black box.

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The sharp Sobolev inequality of Aubin and Talenti plays a critical role in many important theorems in geometric analysis, including the Yamabe problem. Using the co-area formula, it is easy to reduce the proof to proving the inequality for functions on $R^n$ that are a function of the distance to the origin only. This is a $1$-dimensional inequality. But the proofs of this 1-d inequality given by Aubin and Talenti are extremely hard to follow. At least one of them simply cites a paper by Bliss that uses techniques of calculus of variations that I find rather obscure. For this reason, I believe very few people who have used and cited the Aubin-Talenti inequality have ever understood its full proof.

The situation, however, has improved. For those who know the details of the construction of the so-called Brenier map in optimal transportation, there is a full proof of the Aubin-Talenti inequality in a beautiful paper by Cordero-Nazaret-Villani.

For those who do not want to learn the full details of the Brenier map, my collaborators and I have included our paper titled "Sharp Affine $L_p$ Sobolev Inequalities" the full details of the Cordero-Nazaret-Villani proof applied to the 1-dimensional Bliss inequality. In this case, the Brenier map can be constructed using only the fundamental theorem of calculus.

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Determinacy of Borel Games seems like a good example of this.

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Fixed point theorems (such as Brouwer and Kakutani's) are very frequently invoked, specially in Econ. I am not sure how many people are familiar with the proofs. There are many nice proofs available, by the way.

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I've got to put in 2c for ergodic theory: the Multiplicative Ergodic Theorem is widely quoted, but locating a complete proof is hard.

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I think the solution to Hilberts 5th problem is an example. For a while Gromov's polynomial growth theorem was an example because the proof invoked Hilberts 5th.

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This is a good example (I learned Gromov's proof as a student, but not Montgomery-Zippin/Gleason), but I'm not sure how many applications it's had. Recently, though, Green and Tao have had to generalize Montgomery-Zippin for applications, but they've had to delve into the details of the proof, so maybe the situation will be rectified. –  Ian Agol Jun 14 '12 at 17:27
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The Borel isomorphism theorem says that any two Polish (complete and separable metrizable) spaces endowed with their Borel $\sigma$-algebra are isomorphic as measurable spaces if and only if they have the same cardiality and this cardinality is either countable or the cardinality of the continuum.

The result is extremely useful and widely applied in probability theory. It allows one to prove many results for general Polish spaces by proving them for the real line or the unit interval. The proof is actually not that hard, but somewhat messy and gives little useable insight for those not working in descriptive set theory.

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Fundamental lemma (Langlands program) which Ngô Bảo Châu proved and got the Fields medal in 2010.

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Chevalley's theorem: any algebraic group is the extension of a linear algebraic group by an abelian variety.

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In several complex variables it is often desired to be able to solve the $\bar\partial$ equation. The standard tool for that is Hörmander's $L^2$ method and though I suppose that most who use it have at some point read at least a sketch of the proof, most probably aren't familiar with the tedious details that go into the proof.

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The Cohen-Structure theorem in commutative algebra (classifying complete local rings in some sense).

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One of the seminal results in random matrix theory is that in the edge scaling limit, the largest eigenvalue of random Hermitian matrices is the Tracy-Widom distribution. The original proof by Tracy and Widom is full of so many unintuitive technical details that most people who cite it don't understand it. (Or so I've been told).

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The proof for Hilbert's tenth problem, that is, that there is no algorithm to solve general Diophantiane equations.

Benjamin Steinberg has mentioned this above in a comment. I believe the proof is complicated.

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Does this really qualify? Benjamin writes: "I mean a theorem that is often applied". How can you apply a Theorem which says that you cannot solve some general equation? In specific situations, you might be able to do so (perhaps even via algebraic stacks mathoverflow.net/questions/96957). –  Martin Brandenburg Jun 14 '12 at 10:13
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@Martin: The MRDP theorem says much more, namely that every r.e. set is Diophantine, and it (and its formalized versions in fragments of arithmetic) does in fact have useful applications in logic. –  Emil Jeřábek Jun 14 '12 at 11:01
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In a similar vein, the word problem for groups? –  Jonny Evans Jun 14 '12 at 11:48
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It is quite common in group theory to reduce algorithmic problems involving nilpotent groups to Hilberts 10th –  Benjamin Steinberg Jun 14 '12 at 23:49
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Many people apply the theorem $1+1=2$, but how many understand in detail the proof given by Russell and Whitehead in Principia Mathematica? Well, I suppose there are other proofs available....

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Does "by definition of the interpretation o the symbol $2$." counts as a proof? –  Asaf Karagila Jun 15 '12 at 16:32
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Slightly debatable, but my impression is that the fact that injectivity, Property (P), and hyperfiniteness define the same class of von Neumann algebras is used by many people without feeling the need to learn the proofs.

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What about Carleson's theorem that Fourier series of $L^2$ functions converge almost everywhere? I don't read the right literature to see whether this is frequently invoked, but it seems like a useful tool to have.

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I have a feeling that it isn't used that much per se, but rather is there to set the limits/scope of various theorems, and to motivate generalizations. However, I'm not really in the right circles to make an adequate judgment –  Yemon Choi Jun 17 '12 at 9:10
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Saharon Shelah has a series of results he actually calls "black boxes," and uses accordingly (see his paper, "Black Boxes," http://arxiv.org/abs/0812.0656); my understanding is that these are Diamond-like theorems that are provable in ZFC.

(Diamond, for clarification, is a sort of guessing principle: it asserts that there exists a single sequence $(A_\alpha)_{\alpha\in\omega_1}$ such that $A_\alpha\subseteq\alpha$ such that, for any $A\subseteq \omega_1$, the set $$\lbrace \alpha: A_\alpha=A\cap\alpha\rbrace$$ is "large" (specifically, stationary - intersects every closed unbounded subset of $\omega_1$). This principle is not provable in ZFC; it follows from $V=L$ and implies $CH$, but both of these implications are strict. My understanding, which is quite limited, is that Diamond is used in constructions of $\omega_1$-sized structures where one needs to "guess correctly" stationarily often, and that Shelah developed the black boxes to perform many of these same constructions in ZFC alone.)

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The Open Mapping Theorem (known as Banach-Schauder Theorem) is used daily by zillions of analysts. But its proof is far from trivial and is often overlooked by users. It is not just a straightforward consequence of Baire's Theorem.

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I think you are overestimating the difficulty of its proof. While it is not trivial, it is something standard and covered without any problem in a functional analysis course. If the proof is overlooked, that's probably usually because of, well, oversight, not because it is too complicated to be understood by the ordinary analyst. –  Michal Kotowski Jun 14 '12 at 16:49
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Two results I have seen used without proof in undergraduate lectures were:

In both cases the proofs are not long, but were deemed not useful enough to be taught. I am wondering whether this is special to the courses I had or generally common.

Also, various courses on graph theory use some versions of the Jordan curve theorem; even the ones not requiring analysis (speaking of piecewise linear paths) are usually not proven. And several analysis courses don't prove the basic properties of real numbers, instead treating them as axioms.

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I don't think that Tychonoff is a good example. After all, the professor surely knew the proof(s). He just chose not to show them to the students. –  Felix Goldberg Jun 15 '12 at 9:10
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How about Haynes Miller's theorem resolving the Sullivan conjecture?

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The fact that every von Neumann algebra is the direct integral of factors. Every operator algebraist knows this, and could probably more or less explain the proof, but the details are tedious (and kind of useless in practice.) There are similar facts about decomposition of non-singular actions into ergodic ones and representation into irreducible representations.

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Does FLT count?

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Maybe the modularity theorem counts, but I wouldn't say FLT does. –  Steven Gubkin Jun 13 '12 at 21:32
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Dirk, the reason FLT doesn't count is that it is almost never used. –  Joël Jun 14 '12 at 23:14
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