# Each mathematician has only a few tricks

The question "Every mathematician has only a few tricks" originally had approximately the title of my question here, but originally admitted an interpretation asking for a small collection of tricks used by all mathematicians. That question now has many answers fitting this "there exist a small set of tricks used by all mathematicians" interpretation. I find that swapping the quantifiers gives a better question. I.e. I am more interested in hearing about the small collections of tricks of individual mathematicians. Pointing back to the other question above, and Rota's article, what are the few tricks of Erdős, or of Hilbert?

Question: What are the few tricks of some individual mathematicians?

Of course, as the comment in the earlier question quips, a mathematician never reveals tricks...but one can hope. In your answers, please include the name of the mathematician, and their few tricks...perhaps some cool places where the tricks are used, i.e. some "greatest hits" applications of the tricks.

Note, I don't think that knowing these tricks can make you into Erdős or Hilbert, but a long time ago a friend told me that a talented mathematician he knew would approach research problems by asking himself how other mathematicians would attack the problem. This is sort of like writing in another author's style, which can be a useful exercise. Wouldn't it be neat to be able to ask yourself "How would Hilbert have attacked this problem?"

MO is a good place to collect these, because it often takes extended reading (as intimated by Rota) to realize the few tricks used by a certain mathematician. As a community, we may be able to do this.

• @LSpice: I'm totally cool if this gets deleted or closed down. My reason for posting is because I was quite disappointed that the other question ended up interpreted differently from this one. If somehow the answers over there collect the info here, let's close this one down. Jun 16, 2020 at 14:42
• I'd really like to know what Hilbert's tricks were, if there is any truth to Rota's comment. I hope this question stays open. Jun 16, 2020 at 16:21
• I much prefer this question to the one which looks similar but is (IMO) based on a misreading of what Rota wrote; so I have voted to close the other one, and if I could vote to keep this one open I would do so. (Also: hi Jon) Jun 16, 2020 at 20:51
• @LSpice Well I think that the other question is actively inviting answers that do not achieve what Jon is asking for here. I mean, we've already had generic answers to the other question like "interchange the order of summation" or "the Cauchy--Schwarz inequality", and TBH I foresee the quality of answers over there going down rapidly, as every random user goes "oh hai what about this trick I saw" Jun 16, 2020 at 23:18
• Not sure it fits the question, but one demoralizing experience is to be pleased at proving a "new" result, and then to discover that you proved the same result 30 years earlier, and furthermore the earlier proof was much better than the "new" one. Jun 19, 2020 at 12:12

The question is worded in a way that seems to imply we might speak of other mathematician's tricks, but I'm not sure I know the tricks of even my closest collaborators, except by osmosis; so I hope it's OK if I specify my own "one weird trick". The entirety of my research centres around the idea that, if $$\chi$$ is a non-trivial character of a compact group $$K$$ (understood either in the sense of "homomorphism to $$\mathbb C^\times$$", or the more general sense of $$k \mapsto \operatorname{tr} \pi(k)$$ for a non-trivial, irreducible representation $$\pi$$ of $$K$$), then $$\int_K \chi(k)\mathrm dk$$ equals $$0$$.

It's amazing the mileage you can get out of this; it usually arises for me when combining Frobenius formula with the first-order approximation in Campbell–Baker–Hausdorff. Combining it with the second-order approximation in CBH gives exponential sums, which in my field we call Gauss sums although that seems to intersect only loosely with how number theorists think of the matter. Curiously, I have never found an application for the third-order approximation.

• This is exactly the kind of thing I'm looking for, and I think it is delightful to see things like this. It feels like hearing this from you over coffee at a chalkboard between talks at a conference. Thanks! Jun 16, 2020 at 14:44
• Btw, @LSpice, I only phrased the question to ask for other mathematician's tricks because "a mathematician never reveals his tricks". Your answer proves this wrong. Thanks again! I hope to see more of these autobiographical ones. Jun 18, 2020 at 10:33

In an effort to get the ball rolling, and to illustrate why I think several answers on the other question don't really work as answers to this one, let me offer an attempt which I think is in the spirit that Jon intended — although I'm too rusty on the details to provide a proper analysis/explanation/justification.

The late Charles Read was (in)famous for constructing counterexamples in functional analysis, specifically in the world of Banach spaces and then later in the world of Banach algebras. While I don't think Rota's phrase "only a few tricks" does justice to Charles (or indeed was ever meant as being particularly accurate, given Rota's fondness for the soundbite), anyone who's had to study some of Charles's papers in detail will have noticed two themes that recur throughout his work.

1. "very rapidly" increasing sequences, which somehow encode the intuition that one builds a counterexample in stages, and in between each stage you need to go "far enough towards infinity to avoid intefering with what you did previously". These come up in his construction of an operator on $$\ell_1$$ with no non-trivial closed invariant subspaces, but if memory serves correctly they also turned up in the Loy–Read–Runde–Willis paper Amenable and weakly amenable Banach algebras with compact multiplication on constructing commutative radical amenable algebras with various seemingly opposing properties, and also came up in one of his later papers on Frechet algebras. Obviously the notion of separating out building blocks of moderately growing size along a lacunary sequence is an ancient one, but for reasons that I confess I don't fully understand, Charles was able to push this idea much further, usually using combinatorial arguments to keep control of the "localized construction at each stage" so that a sufficiently fast growing sequence would separate them out.

2. When $$N$$ is large "or infinite", the algebra of upper-triangular $$N\times N$$ matrices has a very large (Jacobson) radical, and so looks very different from Banach algebras such as $$L^1(G)$$ or $${\rm C}^\ast$$-algebras which had tended to drive a lot of (over-)optimistic conjectures. There were several papers that seemed, underneath the formidable technical details, to have in mind this mental image: this is explicit in his "Commutative, radical amenable Banach algebras" paper, and implicit in his paper with Ghlaio Irregular abelian semigroups with weakly amenable semigroup algebra that constructs commutative semigroups which are far from being groups yet whose convolution algebras are weakly amenable. My point is that Charles did not just view the fact at the start of this paragraph as a known result to be quoted or used as a black box, he seemed to have a deep appreciation of how to use "identity + strictly upper triangular = invertible, albeit with a large inverse" as a guiding principle in his constructions.

There have been very few papers which seek to explain what is going on in Charles's constructions, either in an expository sense or in an "extend or refine" sense. Two that come to mind are: S. Grivaux and M. Roginskaya's paper A general approach to Read's type constructions of operators without non-trivial invariant closed subspaces; or Chapter 5 of R. Skillicorn's PhD thesis Discontinuous homomorphisms from Banach algebras of operators

(This answer is difficult to write because I feel conscious that I've only managed a very superficial account of what is going on in the papers I refer to. Improvements and corrections would be very welcome.)

• Thank you, Yemon! I hope the ball will keep rolling. I look forward to more answers like this one. I think it is ideal that such an answer invites clarification/correction and simultaneously serves as a kind of "book review" for a favorite theme/trick of the mathematician in question. This answer and the other by @LSpice are good prototype answers to this question. Jun 17, 2020 at 0:25
• A superficial account, in this sense of a million-mile overview, is surely what a question about tricks looks for—if you want the details, then read the paper! Jun 17, 2020 at 17:10

“Most mathematicians know one method. For example, Norbert Wiener had mastered Fourier transforms. Some mathematicians have mastered two methods and might really impress someone who knows only one of them. John von Neumann had mastered three methods: 1) A facility for the symbolic manipulation of linear operators, 2) An intuitive feeling for the logical structure of any new mathematical theory; and 3) An intuitive feeling for the combinatorial superstructure of new theories.” - Ulam

So I guess that covers Wiener and von Neumann

• I'm pleased that two of von Neumann's three methods are of the form "An intuitive feeling ...", because this agrees with my opinion that intuitive feelings (when not misleading) are extremely valuable --- and correspondingly hard to acquire. Jun 26, 2020 at 21:36

It feels a bit presumptuous to talk about another mathematician's favorite tools. However, there is something known as Uhlenbeck's trick, which definitely deserves mentioning.

One recurring theme in Karen Uhlenbeck's work is to use gauges in clever ways which make analysis tractable. For example, Terry Tao wrote a blog post about a deep result about connections with small curvature that she proved by combining the right choice of gauge with the continuity method.

The named version of this trick uses this idea in the context of Ricci flow. In simple terms, one uses an orthonormal frame which evolves in time and where the curvature evolution equations greatly simplifies. From a more conceptual standpoint, the idea is to consider an vector bundle $$V$$ which is isometric to the tangent bundle $$TM$$ and has a fixed metric $$h$$. Then, the Ricci flow acts to evolve the isometry between $$V$$ and $$TM$$. Although this is conceptually more complicated, the use of the fixed metric $$h$$ simplifies the evolution equations and allows one to find invariant curvature conditions, which plays an essential role in the analysis.

• You are right! I have removed the word "favorite" from the body of the question. The idea is, though, to associate mathematicians to the tools they tend to use and the way they used these tools. It would be very funny to claim that a trick was among someone's favorites and for that person to find out about it via that claim. Thank you for the nice answer. Jun 18, 2020 at 17:54
• Thanks. As far as named tricks go with the Ricci flow, there's also the Deturck trick. I'm not sure how that fits into his larger body of work so I didn't mention it in the answer. Jun 19, 2020 at 19:09
• As far as I'm aware, the DeTurck trick was something particular to Ricci flow, I think the trick has found other uses since then, although I can't remember if its used elsewhere by him (wouldn't surprise me if it was though). Aug 16, 2020 at 18:12
• My understanding of the Deturck trick is that you conjugate Ricci flow by a time-dependent diffeomorphism which produces a parabolic flow (and so bypasses the need for Hamilton's technical proof of existence). If Deturck used this type of idea elsewhere, it would be a good answer for this question. Aug 16, 2020 at 21:33

I have two tricks: Dehn filling and drilling. I've used the former to study subgroup separability, as a technical trick to reduce the proof of tameness of Kleinian groups in the cusped case to the non-cusped case, to produce non-Haken 3-manifolds, as well as study exceptional (non-hyperbolic) Dehn fillings on a cusped manifold. I've used drilling also in the proof of tameness, to relate the volume of closed hyperbolic manifolds to cusped ones, and in the solution of Simon's conjecture about epimorphisms between knot groups.

As you might guess, these are really the same trick (one is the inverse operation of the other), but I like to think of them as two ;).

When I was an undergraduate, I attended a talk by Peter Lax in Budapest. He had recently been awarded the Abel Prize, but attributed all his success to "integration by parts." It seems he has said this publicly a few times.

• +1 though since Lax first allegedly said it, it has become rather cliche that this trick is the trick for many people who work in partial differential equations or harmonic analysis. :) Jun 19, 2020 at 17:15
• This is from the obituary in Notices of the AMS: In 1948 Laurent Schwartz visited Sweden to present his distributions to the local mathematicians. He had the opportunity of conversing with Marcel Riesz. Having written on the blackboard the integration-by-parts formula to explain the idea of a weak derivative, he was interrupted by Riesz saying, “I hope you have found something else in your life.” Jun 20, 2020 at 13:23

I like the idea of trying to recognize a mathematician by their tricks. It reminded me of the Brachistochrone problem, posed by Johann Bernoulli and solved by five mathematicians, including an anonymous solution by Newton. This is the source of Bernoulli's famous quote "tanquam ex ungue leonem," Latin for "we know the lion by his paw." What was it that made Newton's approach so immediately recognizable? It was his use of the Calculus of Variations, which he had used ten years earlier to solve the Minimal Resistance Problem. This approach uses in a fundamental way: intuition from physics, approximating infinitesimal curves by infinitesimal lines, and the use of truncated power series expansions. I'd say those tricks were quintessentially Newton's.

• Isn't it "… by his claw"? Jun 18, 2020 at 14:42
• I checked this carefully, but know little about Latin. From what I understand, it does not literally translate to either phrase (throw it in Google translate if you don't believe me). I learned "paw" but a few sources I found said "claw". Probably depends who you heard the story from. Jun 18, 2020 at 15:59

In my field (symmetric functions and representation theory) there are a few tricks that some people are quite notorious for.

• S. Assaf - Introduce new families of polynomials/(quasi)symmetric functions, and use dual equivalence.
• P. Brändén - Generalize real-rootedness to the notion of stability.
• A. Garsia - Introduce new operators acting on symmetric functions.
• M. Haiman - Use super-hardcore algebra stuff to prove things about symmetric functions.
• C. Krattenthaler - Compute a determinant.
• D. Zeilberger - Use computer algebra (the WZ-algorithm in particular) and let S.B Ekhad do all the actual work!
• I love "compute a determinant". Jun 17, 2020 at 19:44
• A. Zelevinsky: show that there is only One Ring. Jun 17, 2020 at 21:08
• D. Knuth: rewrite all algorithms in your own assembly language. Jun 17, 2020 at 21:10
• A. Vershik, A. Okounkov: find enough $\mathfrak{sl}_2$-triples in the algebra. Jun 17, 2020 at 21:12
• I. G. Macdonald: introduce 9 new families of polynomials in one paper. Jun 17, 2020 at 21:16

Erdős' trick is discussed at length in Gowers' classic essay Two Cultures of Mathematics, where he describes it as follows:

If one is trying to maximize the size of some structure under certain constraints, and if the constraints seem to force the extremal examples to be spread about in a uniform sort of way, then choosing an example randomly is likely to give a good answer.

This is often combined with the following trick introduced by Shanon:

The expected value of a random variable is between its minimum and its maximum. Therefore you can prove lower bounds on the largest possible value of a function on a set of objects by examining the expected value of that function a random object.

One example of combining these techniques is the following well-known result:

Theorem: Every 3-SAT instance has an assignment of variables that satisfies 7/8ths of the clauses.

Proof: A random assignment of values satisfies 7/8ths of the clauses in expectation, as any particular clause is only false if all of its constituent variables are false.

We can even covert this into an efficient, deterministic, constructive proof! Let $$S$$ be the random variable that returns the number of clauses satisfied by a random assignment. Set the value of $$x_0$$ to $$0$$ (resp. $$1$$) and call the restricted version of $$S$$ that satisfies this condition $$S_0$$ (resp. $$S_1$$). Then $$\frac{7}{8}=\mathbb{E}S = \frac{1}{2}\mathbb{E}S_0 + \frac{1}{2}\mathbb{E}S_1$$, so at least one of the expected values on the right are $$\geq 7/8$$. That one tells you the correct value for $$x_0$$, and then now iterate.

• Cool! Thank you! Jun 19, 2020 at 18:00
• Erdős' trick is expanded upon at book length in "The Probabilistic Method" by Alon and Spencer. Feb 24 at 17:24
• Although Erdős largely pioneered this "trick" and used it effectively many times, this is not an example of a mathematician having only a few tricks. Erdős is rightly regarded as great because he had many wide-ranging "tricks". Feb 25 at 2:50

Tao has recently submitted a preprint on exactly this topic in the case of the mathematician Jean Bourgain. The tricks in question are quantification of qualitative estimates, dyadic pigeonholing, random translations, and metric entropy and concentration of measure. As you say, he points out that knowing these tricks does not automatically give you the intellectual firepower of Bourgain, but that they are very useful nonetheless.

• This is really nice! It complements the answers to another MO question about Bourgain. Sep 17, 2020 at 21:29
• Not only this, but the article you link to gives a perfect answer to my question here, interpreting the Rota quotation as I believe it was meant to be interpreted... Jul 28, 2021 at 19:44

Not me, but Donald Ervin Knuth:

Use clever notation! Especially for sums, recurrences, binomials, etc. he developed very useful variations (Concrete Mathematics [Graham, Knuth, Patashnik], The Art of Computer Programming [Knuth])

The notations he proposes are clear, and, more importantly, lead to an amazing amount of intuition, which wouldn't be possible otherwise.

• Unfortunately one person's clear, intuitive notation is another person's awkward mess—or even, as I discovered when re-visiting my own notation later, the same person's awkward mess when the purpose to which it is to be applied shifts ever so slightly. Aug 8, 2020 at 1:20
• Ervin, not Edwin. Gerhard "Not Talking Evil Twin Here" Paseman, 2020.08.08. Aug 8, 2020 at 11:05
• @GerhardPaseman Fixed. What an embarrassing mistake. Aug 15, 2020 at 10:19
• @LSpice While I certainly mostly agree with your point of view, I kind of feel Knuth is a bit of an exception in that regard: He (usually) doesn't re-invent the wheel, but 'fixes' the exact problems you mention in existing notation by giving it a small, but clever, twist. - But it always comes down to personal taste and the problem you are trying to solve. Aug 15, 2020 at 10:27

I want to mention a trick of Gilles Pisier. This is an extrapolation method. Suppose you have some kind of inequality for some $$L^p$$ space and that you want to get a reverse Holder type inequality for $$q. Using this he has done many interesting work in Sidon sets, Grothedieck inequality and noncommutative Khintchine's inequality. The trick is originally attributed to Rudin's famous paper "Trignometric Series with Gaps".

Here is Jon's reply and some more explanations. In the paper "Trignometric Series with Gaps", Rudin deals with the following kind of sets. Let $$0 A set $$E\subseteq \mathbb Z$$ is of type $$(r,s)$$ if $$\|f\|_s\leq B\|f\|_r$$ for all trignometric polynomials in $$\mathbb T$$ with Fourier coefficients of $$f$$ supported on $$E.$$ Rudin proves that for $$0 $$E$$ is of type $$(r,t)$$ if and only if it is of type $$(s,t).$$ The proof uses a reverse Holder kind of inequality. It is an extrapolation trick, i.e. knowing something for $$(s,t)$$, one extrapolates to $$(r,t).$$ The same kind of trick was used for proving noncommutative Khintchine inequality (https://arxiv.org/abs/1412.0222) for $$p<1$$. However, in every case the trick involves some new technical difficulties but the philosophy is the same. Pisier used same kind of trick to obtain a new upper bound of complex Grothendieck constant (https://www.sciencedirect.com/science/article/pii/0022123678900381). There are many other instances. One can look carefully into his papers and will see that many times he used this trick.

• Thank you for the extended answer!! Jun 20, 2020 at 14:12

Gabe's answer, about Uhlenbeck's trick, reminded me of the Rabinowitsch trick in algebraic geometry. However, I don't know if Rabinowitsch used this trick in other work, or if it was indicative of his approach to mathematics. Good thing this is community wiki! I encourage anyone who knows more to edit with more details.

Characterizing a class of integers sharing some property $$P$$ by defining an arithmetic function taking a single value $$k_{P}$$ at those integers and then give an equivalent of this arithmetic function.

Finding properties of an object that are invariant under the action of some natural involution.

Saying mathematicians have only "a few tricks" makes mathematicians seem rather limited. But I recall some saying that great philosophers are engaged with only one big question. Perhaps this is true for all fields, after all there is that old adage which underlines this: jack of all trades, master of none.

I recall reading somewhere that Ramanujan had a 'master technique'. According to Wikipedia this was the Mellin transformation of a function expressed as a power series.

Feynman in one of his popular books mentioned that he could often do integrals that his colleagues couldn't because he knew how to differentiate under the integral sign. Some people have begun to call it Feynman's trick.