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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

The existence of resolution of singularities in characteristic zero is certainly used by many more people than those who know the details of its proofs, especially the original one.

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There are many papers by H. Hauser whose message is "You can understand Hironaka's proof!". There is even a game-theoretic interpretation. Very recommended. –  Martin Brandenburg Jun 14 '12 at 8:15
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There are also now books (one by Kollar and another by Cutkosky) that aim to present the proof at a graduate student level. I think they don't prove the most general/detailed statements from Hironaka's original paper, though. –  Dan Ramras Jun 15 '12 at 0:34
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The Cohen-Structure theorem in commutative algebra (classifying complete local rings in some sense).

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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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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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My immediate thought upon seeing the question, and, I believe, one of the biggest examples of this phenomenon, is:

Class Field Theory

Almost anyone working in algebraic number theory uses the main results of class field theory regularly. However, even if many people have sat through a course going through the proofs of the theorems, very few people remember the proofs, and even fewer use them.

I recall hearing advice from various mathematicians that the most important thing is to learn the statements of class field theory, but not the proofs.

See, for example, this quote from the Syllabus to Brian Conrad's course on class field theory:

While it is somewhat instructive to know what goes into the proofs of the main theorems (e.g., to see what obstacles prevent the proofs from being entirely constructive), it cannot be said that the grungy details of these proofs are particularly relevant to using the theory in practice. Thus, in the first half of the course we will emphasize an understanding of the statements of the main results (in their many different forms) and will not place much emphasis on how the main theorems are proven; precise references will be given for those who wish to read the details of the proofs of the main theorems. Once we have spent some time digesting what class field theory tells us, we will study some applications of the theory, such as in the context of imaginary quadratic fields and abelian coverings of algebraic curves.

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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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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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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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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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The Feit–Thompson Theorem stating that every finite group of odd order is solvable.

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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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Jordan's curve theorem is used as a blackbox.

This topology theorem states that a looped continuous path in the plane partitions the points of the plane, such that any continuous path going from a point in one partition to a point in the other intersects the loop.

There seem to be a lot of theorems in calculus of which I don't fully understand the proof, though some of this shows my ignorance of calculus. Jordan's theorem seem to be an extreme example though. Let me list some other examples.

  • the existance and basic properties of the Lebesgue measure and infinite product measures
  • the fact that a Wiener process is almost surely everywhere continuous (mentioned below as a separate answer by weakstar)
  • the fact that the roots of a complex polynomial (or the eigenvalues of a complex matrix) are continuous in the coefficients (though I should learn the proof for this because the more precise statements on how well conditioned the roots are on the coefficients is useful)
  • the spectral theorem about linear maps on a possibly infinite-dimensional Hilbert-space
  • the proof that a convex function (from reals to reals) is always continuous everywhere and has a left and right derivative everywhere (Update: okay, remove this last one because Ian Morris gave a simple proof below. I seemed to remember it was more difficult than that. Thanks, Ian.)
  • Rademacher's theorem: every Lipschitz function from an open subset of $ \mathbb{R}^m $ to $ \mathbb{R}^n $ is differentiable almost everywhere. (Added on Paul Siegel's suggestion. For some reason I haven't heared of this theorem before, but it sure sounds useful.)
  • Lebesgue's criterium which claims that a bounded function from reals to reals is Riemann-integrable iff it's continuous almost everywhere. (The proof is elementary and doesn't require any ideas, but it's laborous.)
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This example is not really what I want because all basic algebraic topology books do it. –  Benjamin Steinberg Jun 13 '12 at 22:59
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Most of those are genuinely laborious proofs, but the one about convex functions can be done in a few lines. A convex function clearly has at most two intervals of monotonicity, which implies that the left and right limits at each point exist. If they aren't the same for some point then we can find a chord between two points of the graph close to the discontinuity which passes below the graph (on the left if the jump is downwards, or to the right if it is upwards) contradicting convexity. Differentiability is obtained by showing that (f(x+r)−f(x))/r is monotone in r. –  Ian Morris Jun 14 '12 at 13:12
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Perhaps you could replace your fifth example with Rademacher's theorem: every Lipschitz function from an open subset of $\mathbb{R}^n$ to $\mathbb{R}^m$ is differentiable almost everywhere. This is a more serious result which people use all the time, and I'm not sure everyone really knows the proof (though maybe I should speak for myself). –  Paul Siegel Jun 14 '12 at 23:39
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Someone mentioned existence and uniqueness of Haar measure on a locally compact topological group. But if one uses the Riesz representation theorem and Tychonoff, the standard proof is not so long or hard, and may even be considered conceptual. For example a clear proof is in Bourbaki's Integration, and in Principles of Harmonic Analysis [by Deitmar and Echterhoff].

I think

the Riesz representation theorem (about the dual of $C_c(X)$)

is more often used as a Blackbox theorem. Of course this is a main result in analysis, and many standard books (Rudin, Folland, Appendix of Conway's Functional analysis) have a proof, but they are all long and technical, and in my opinion very difficult to remember. See also Remark 4 in these wonderful notes by Terry Tao.

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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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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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Perhaps Atiya-Singer's index theorem can qualify. Another candidate is Gromov's h-principle.

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The situation with the h-principle is certainly complicated by the fact that there are multiple h-principles with different proofs. –  Igor Khavkine Jun 18 '12 at 20:47
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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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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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The existence of Hilbert and Quot schemes. These are arguably the most important objects in moduli/deformation theory but the proof of their existence is almost never even presented in books on the topic, let alone needed or used. All the properties and applications follow formally so the existence is used as a black box.

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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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How about Haynes Miller's theorem resolving the Sullivan conjecture?

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C*-algebra theory has a number of good examples of this.

  • Voiculescu's theorem: an ample representation of a C*-algebra essentially absorbs any nondegenerate representation
  • Kasparov's technical theorem: if anybody really cares, I'll repeat the statement. The point is that anybody who works with bivariant K-theory uses this result ALL THE TIME, e.g. for excision or the existence of Kasparov products.
  • Stinespring's theorem: any completely positive map into $B(H)$ dilates to a representation

I have been using Voicalescu's theorem and KTT for a about a year or so longer than I knew the proofs. I probably still wouldn't know the proofs if it hadn't become necessary. Stinespring's theorem is probably better known among the people who use it because it's not so difficult, but it could be tempting to use it as a black box.

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Nice answers. An addendum/afterthought: it probably isn't used very often by practising operator algebraists, but the equivalence of nuclearity and amenability for C*-algebras gets invoked a lot of the time by people in Banach algebras, and I rather doubt many of them have actually worked through all the details. –  Yemon Choi Jun 15 '12 at 0:26
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I think differential topology has dozens of these results. Here are some examples that immediately come to mind:

  • The tubular neighborhood theorem: every submanifold $N$ of a manifold $M$ has an open neighborhood which is diffeomorphic to the total space of the normal bundle of $N$
  • The fundamental theorem of Morse theory: if $f: M \to \mathbb{R}$ is a Morse function and $[a,b]$ is an interval which contains no critical values of $f$ then the set of all points where $f \leq a$ is a deformation retract of the set of all points where $f \leq b$
  • Every continuous isomorphism of smooth vector bundles is homotopic to a smooth isomorphism (and other such "continuous equivalence = smooth equivalence" results)

Probably most topologists know the basic ideas behind the proofs of these results, but I think many would be hard-pressed to actually write down a complete argument. I say this with confidence because I know of several textbooks by good authors that have proofs which are either wrong or sketchy on some details.

There are also some results with standard proofs that are widely known, but I think considerably more people use the results than know the proofs:

  • De Rham's theorem: the De Rham cohomology groups of a manifold are isomorphic to the singular cohomology groups with real coefficients
  • The Hodge theorem: every De Rham cohomology class on a Riemannian manifold has a harmonic representative
  • Whitehead's result that every smooth manifold has a unique PL structure
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Definitely, definitely true. Even more so for "the second part" of "the fundamental theorem of Morse Theory", that passing a critical point attaches a handle: mathoverflow.net/questions/70248/… –  Daniel Moskovich Jun 15 '12 at 0:24
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This doesn't quite seem to be in the spirit of what the original questioner was going for. This is more like, "Mathematical facts that people think that they understand but are actually a bit trickier than they think," which is interesting, but perhaps deserves its own thread. –  Dan Lee Jun 15 '12 at 20:36
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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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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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For a long time, the Littlewood-Richardson rule has been a black box. See van Leeuwen's wonderful article for its history (and a rather involved, even if enlightening proof). This really changed with Stembridge's 2-pages long slick (although far from straightforward!) proof (2002) and Gasharov's 3-pages long proof (1998). (I have read Stembridge and can vouch for its good exposition; it's not short by virtue of being unreadable, but short by virtue of being short. I have not yet read Gasharov, and I am in the middle of van Leeuwen.)

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I think the main statements of the MMP (Minimal Model Program ) in algebraic geometry qualify for this.

It will even become more of a black box in the future, as people will understand better how to apply it.

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Can you be a little more specific? do you refer to minimal models in algebraic topology? –  Gil Kalai Jun 14 '12 at 11:40
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Faltings' almost purity theorem. The proof given, for the smooth case, in $p$-adic Hodge theory has some problems, and the proof of the general case in the Asterisque paper Almost Étale Extensions is completely unreadable (at least to me) and also contains some mistakes. We now finally have a very good proof (by Peter Scholze), but the almost purity theorem has been used as a black box for years.

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

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