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Questions that are about research in mathematics, or about the job of a research mathematician, without being mathematical problems or statements in the strictest sense. Do not use this tag for easy or supposedly easy mathematical questions.

208 votes
72 answers
51k views

What are your favorite instructional counterexamples?

Related: question #879, Most interesting mathematics mistake. But the intent of this question is more pedagogical. In many branches of mathematics, it seems to me that a good counterexample can be w …
23 votes
0 answers
651 views

Which proofs of the fundamental theorem of algebra are "essentially the same" vs. "essential...

The classic MO thread Ways to prove the fundamental theorem of algebra contains $60$ proofs of FTA, and I'm sure there are many more in the literature. It would be nice to have some way to organize th …
33 votes

Mathematicians who were late learners?-list

Sophus Lie didn't become interested in mathematics until after university, and before then didn't seem to have shown significant aptitude for it.
Martin Sleziak's user avatar
99 votes

Your favorite surprising connections in mathematics

From an essay of Arnol'd: Jacobi noted, as mathematics' most fascinating property, that in it one and the same function controls both the presentations of a whole number as a sum of four squares and t …
Peter LeFanu Lumsdaine's user avatar
4 votes

Why are characters so well-behaved?

For what it's worth, I had exactly the same question as you and worked out the proof that Noah sketches in some detail at The orthogonality relations for representations of finite groups, although I d …
LSpice's user avatar
  • 12.9k
28 votes

Dimension leaps

The Leech lattice. At least, in the sense that 24 is one of the only dimensions where we know what the densest lattice packing looks like. As usual, John Baez's thoughts. Conway and Sloane is a good …
The Amplitwist's user avatar
47 votes
33 answers
17k views

What are the most overloaded words in mathematics? [closed]

This is community wiki. In each answer, please list one word at the top and below that list as many different meanings of that word in mathematics as you can think of, preferably with links or defini …
52 votes

Theorems with unexpected conclusions

I learned this example from Noam Elkies's excellent article The Klein Quartic in Number Theory. Elkies observes that Siegel's 1968 paper Zum Beweise des Starkschen Satzes, in order to prove its main …
Martin Sleziak's user avatar
12 votes

Theorems with unexpected conclusions

Here's one I was reminded of recently. Recall that a projective plane is a triple $(P, L, I)$ where $P$ is a set of "points," $L$ is a set of "lines," and $I$ is a subset of $P \times L$ describing t …
Martin Sleziak's user avatar
20 votes

What are some examples of colorful language in serious mathematics papers?

I always liked Edward Burger's A Tail of Two Palindromes. It begins as follows: Upon a preliminary perusal, this parable may appear to be about pairs of palindromes, periods, and pitiful alliteration …
Martin Sleziak's user avatar
22 votes

Jokes in the sense of Littlewood: examples?

Let $C(x) = \sum_{n \ge 0} \frac{1}{n+1} {2n \choose n} x^n$ be the generating function for the Catalan numbers. Then $C(x) = \frac{1 - \sqrt{1 - 4x}}{2}$. In particular, $C(1) = \frac{1 - \sqrt{-3} …
David Roberts's user avatar
  • 35.5k
22 votes

Examples of great mathematical writing

Gian-Carlo Rota's On the foundations of combinatorial theory I: Theory of Möbius Functions is an eye-opening gem. The same is true of practically every paper in Gian-Carlo Rota on Combinatorics, so co …
Martin Sleziak's user avatar
195 votes

Most harmful heuristic?

This isn't really a heuristic, but I hate "functions are formulas". For most students it takes a really long time to think of a function as anything other than an algebraic expression, even though nat …
sɪʒɪhɪŋ βɪstɦa kxɐll's user avatar
12 votes

Most memorable titles

Simmons, F. W., When Homogeneous Continua Are Hausdorff Circles (or Yes, We Hausdorff Bananas), Continua, Decompositions, and Manifolds, University of Texas Press (1980) pp. 62-73. I think it's a ref …
Martin Sleziak's user avatar
4 votes

Quick proofs of hard theorems

People say that Hilbert's basis theorem was once proven using pages of explicit computation with polynomials, but now everyone learns Hilbert's beautiful, if non-constructive, proof instead. Regretta …
Martin Sleziak's user avatar

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