122 votes

Every mathematician has only a few tricks

$$ \sum_{i=1}^m\sum_{j=1}^n a_{i,j}=\sum_{j=1}^n\sum_{i=1}^m a_{i,j} $$ (and its variants for other measure spaces). I still get misty-eyed whenever I read something that capitalizes on this trick in ...
106 votes

Small ideas that became big

In a letter to Frobenius, Dedekind made the following curious observation: if we see the multiplication table of a finite group $G$ as a matrix (considering each element of the group as an abstract ...
105 votes
Accepted

Conceptual reason why the sign of a permutation is well-defined?

(This is a variant of Cartier's argument mentioned by Dan Ramras.) Let $X$ be a finite set of size at least $2$. Let $E$ be the set of edges of the complete graph on $X$. The set $D$ of ways of ...
Bjorn Poonen's user avatar
  • 23.6k
88 votes

Every mathematician has only a few tricks

A very useful generic trick: If you can't prove it, make it simpler and prove that instead. An even more useful generic trick: If you can't prove it, make it more complicated and prove that instead!
79 votes

Endless controversy about the correctness of significant papers

(Also mentioned in Oliver Nash's comment) From a February 2017 article in Quanta Magazine called "A fight to fix geometry's foundations" (the original has relevant links in the text): ...in 2012, ...
79 votes

Small ideas that became big

The problem of the seven bridges of Königsberg is surely one of the best-known examples of this. Euler apparently didn't even consider this problem to be mathematical when he solved it, but in doing ...
76 votes

Every mathematician has only a few tricks

In combinatorics: shove it into OEIS, and see what's up. Also, add more parameters! Note: the Macdonald polynomials were introduced by adding more parameters to the Jack and the Hall-Littlewood ...
76 votes

Every mathematician has only a few tricks

Dennis Sullivan used to joke that Mikhail Gromov only knows one thing, the triangle inequality. I would argue that many mathematicians know the triangle inequality but not many are Gromov.
76 votes

Conceptual reason why the sign of a permutation is well-defined?

This is obviously a subjective question, but I teach this in two phases (1) I need to know this fact very early, so I give the quickest definition I know: $$\mathtt{sign}(\sigma) = \frac{\prod_{i<j}...
David E Speyer's user avatar
72 votes

17 camels trick

Slack variables in linear programming. Quote from the link: In an optimization problem, a slack variable is a variable that is added to an inequality constraint to transform it to an equality. ...
68 votes

Small ideas that became big

Cantor's monumental investigation of the infinity started very innocently as a method to understand the uniqueness of the representation of a function by trigonometric series.
64 votes

17 camels trick

Lagrange multipliers: to extremize $f(x,y)$ subject to constraint $g(x,y)=0$, add a variable $\lambda$ and introduce an auxiliary function $$ {\mathcal {L}}(x,y,\lambda )=f(x,y)-\lambda \cdot g(x,...
63 votes

17 camels trick

Maybe this example is too elementary for this site, but I'd say the proof that a closed subset of a compact set is compact itself looks like this. Given an open cover of a $A$ where $A$ is a closed ...
63 votes

Is data science mathematically interesting?

I will stay away from the academic politics of hiring "professors of data science", but if I interpret the question more specifically as "does data science offer problems of mathematical interest", I ...
61 votes

Endless controversy about the correctness of significant papers

The Jordan curve theorem asserts that a simple continuous closed curve separates the plane into two distinct connected open sets. Whether Camille Jordan's original proof is correct or not seems to be ...
61 votes
Accepted

Are there any fields of academic mathematics whose epistemic status as math is controversial within the academic community?

There are some speculative mathematical concepts that come to mind, such as the field of one element or motives, though perhaps these are more classifiable as "potential future mathematics" ...
60 votes
Accepted

Is spherical trigonometry a dead research area?

It is not. As a proof, I will mention three relatively recent papers where I am a co-author: M. Bonk and A. Eremenko, Covering properties of meromorphic functions, negative curvature and spherical ...
Alexandre Eremenko's user avatar
59 votes

Is amateur research in mathematics viable?

This is possible. I have at least two friends who studied mathematics (in the graduate school), did not defend their PhD, and found some jobs not related to mathematics. Still they do research, and ...
57 votes

Each mathematician has only a few tricks

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'...
55 votes

Every mathematician has only a few tricks

Integration by parts has allegedly earned some people big medals.
54 votes

17 camels trick

The AM-GM inequality (as well as other inequalities, for instance the discrete version of Jensen's inequality) can be proved with the following trick due to Cauchy, known as "forward-backward ...
53 votes

17 camels trick

The usual way to prove the Strong Nullstellensatz from the Weak Nullstellensatz is the so-called Rabinowitsch trick. In short, adding an extra variable allows us to apply the "weak" version of the ...
53 votes

What mathematical problems can be attacked using DeepMind's recent mathematical breakthroughs?

This is a bit speculative, and perhaps too challenging for an undergraduate project, but I wonder if an AlphaGeometry type approach might be possible for the task of automatically upper bounding sums ...
Terry Tao's user avatar
  • 108k
51 votes

Fascinating moments: equivalent mathematical discoveries

Consider $n$ evenly spaced points on a circle representing $\mathbb{Z}^n$. Two sets of points with the same multiset of distances between them (measured by the shortest distance around the circle) are ...
51 votes

Conceptual reason why the sign of a permutation is well-defined?

Draw a braid on $n$ strings sloppily, so that no three strings pass through the same point. Convince yourself that the parity of the number of crossings is the same in every other drawing of the ...
Wilberd van der Kallen's user avatar
50 votes
Accepted

Why aren't proceedings from ICM 2014 on mathscinet?

We have had difficulty obtaining the requisite permissions from the publisher. The ICM2014 website has the Legal Disclaimer: "The Seoul ICM Organizing Committee, the legal copyright owner of the ...
Edward Dunne's user avatar
  • 2,443
50 votes

Changes forced by the pandemic

Online seminars Research (and other) seminars have gone virtual. The obvious advantage is, that anyone can attend from basically all over the world. The page https://researchseminars.org/ compiles a ...
50 votes

Situations where “naturally occurring” mathematical objects behave very differently from “typical” ones

Most finite groups empirically are 2-groups (in the sense of being a p-group with $p=2$ not in the other sense of the word). There are a lot of them. Conjecturally almost all finite groups are 2-...
50 votes

What do you do when you're stuck?

Here is an answer which may be math-specific: If you are stuck in some proof of some claim that you believe is true: Add the missing piece as assumption and continue as planned.

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