Each mathematician has only a few tricks

Question

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.

Answer 1

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.

Answer 2

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.)

Answer 3

“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

Answer 4

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 ;).

Answer 5

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. 

Answer 6

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.

Answer 7

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.

Answer 8

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!

Answer 9

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.

Answer 10

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.

Answer 11

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.

Answer 12

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 $q<p$. 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 $0<r<s<\infty.$ 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 $0<r<s<t<\infty,$ $E$ 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.

Answer 13

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.

Answer 14

In all 200+ pages of my category theory notes, there were essentially three tricks I used in proofs:

1. Prove that two arrows are both the arrow induced by a universal property, so they're the same arrow.

2. Reason under the image of a faithful functor between two objects, then conclude that because they're the same under the image of a faithful functor they're equal in the original category.

3. Use diagrammatic coherence conditions to write an arrow in a new representation.

Over the course of hundreds of proofs, I don't think I ever had to do more than these three things.

I would be curious to see a theorem/proof in pure $1$-category theory that uses any method besides these three.

Answer 15

A trick/technique that I like (and used) a lot is the formal geometry approach (after Gelfand-Kazhdan) for passing from a local to a global result.

Let $X$ be a $d$-dimensional manifold. There is a an infinite dimensional manifold $X^{coord}$ whose points are pairs $(x,\varphi)$ , where $x\in X$ and $\varphi$ is a formal coordinate system around $x$ (in other words, $\varphi$ is an isomorphism between the formal neighborhood $\hat{X}_x$ of $x$ in $X$ and the formal neighborhood $\hat{\mathbb{R}}^d_0$ of $0$ in ${\mathbb{R}}^d$.

$X^{coord}\to X$ is a $G_d$ principal bundle, where $G_d$ is the automorphism group of $\hat{\mathbb{R}}^d_0$; hence $G_d$ is actually homotopy equivalent to $GL(d)$. Somehow, working $GL(d)$ equivariantly on $X^{coord}$ amounts to work on $X$.

This is of course not the whole story: $X^{coord}$ is in fact universal among spaces $Y\to X$ satisfying $Y\times_{X}\widehat{X\times X}\simeq Y\times \hat{\mathbb{R}}^d_0$, where $\widehat{X\times X}$ denotes the formal neighborhood of the diagonal in $X\times X$.

This allows to give a precise meaning to the idea that $X$ is obtained as a kind of glueing of all formal neighborhoods of its points.

In derived geometry, these formal geometry methods have been subsumed by techniques involving Simpson's de Rham stack.

Edit : Fedosov used such methods to prove the existence of a star-product on a general symplectic manifold in A simple geometrical construction of deformation quantization (locally, existence is given by the Moyal star-product). Similarly, Kontsevich's proof of his famous formality theorem, in Deformation quantization of Poisson manifolds, goes in two steps: (a) prove a local formula, (b) globalize. The second step is proven using formal geometry methods. But one has to say that (a) is way more difficult than (b) for the formality theorem (whereas in Fedosov's paper, local existence is trivial).

Answer 16

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.

Answer 17

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.

It's also worth pointing out that writers only know 26 letters. But out of that has poured out all our literature. Some trick!

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.

Answer 18

In my personal field, applied optimal transport for PDEs, we often play the following game, so much so that some of my colleagues and I actually call it Brenier's trick (after Yann Brenier): When miniminzing a convex functional $\rho\mapsto F(\rho)$ over some convex subspace (typically the space of probability measures) with linear constraints $L(\rho)=0$, write the constraints as a supremum of linear functionals over auxiliary mutlipliers, and use a convex/concave minmax theorem (Rockafellar, often, or any variant of von Neumann's minmax theorem in infinite dimension) to swap $\inf\sup=\sup\inf$ as \begin{multline*} \inf\limits_\rho\Big\{ F(\rho):\quad L(\rho)=0\Big\}=\inf\limits_{\rho} F(\rho)+ \begin{cases} 0 & \text{if }L(\rho)=0\\ +\infty&\text{else} \end{cases} \\ =\inf\limits_\rho\Big\{ F(\rho)+\sup\limits_\phi \langle L(\rho),\phi\rangle\Big\} =\inf\limits_\rho\sup\limits_\phi F(\rho)+\langle L(\rho),\varphi\rangle \\ =\sup\limits_\phi\inf\limits_\rho F(\rho)+\langle L(\rho),\varphi\rangle. \end{multline*} One can then solve the free optimization problem in $\rho$ and next in $\phi$ to retrieve a lot of significant information about the joint optimizers. For example this can be used to retrieve in just a few lines, and at least heuristically, the right equations for Wasserstein geodesics (a forward continuity equation coupled with a backward Hamilton-Jacobi equation), the so-called Otto's calculus, and many other variants thereof for a whole variety of models. Of course this is nothing but a Lagrange multiplier method for constrained optimization, but it shows up so often in applied optimal transport problems that I thought it would be worth mentioning here.

Answer 19

A trick that is used daily is Zorn's Lemma. Sure, every mathematician knows it, but it certainly helped prove non-trivial propositions and it is in daily use. I would consider it in a list of the Top 3 Tricks of Maths as it is used in such diverse fields as algebra, real analysis, general topology.