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I can't believe no one's mentioned this: Some title containing the words "homotopy" and "symplectic", e.g. this one Pavol Severa arXiv:math/0105080


A natural choice is Riemann's "On the Number of Primes Less Than a Given Magnitude" at only 8 pages long...


In my experience, mathematicians will frequently argue (in general, not just in mathematics) by passing to an extreme case at the beginning. Non-mathematicians (again in my experience) sometimes object to such a mode of argument as invalid or irrelevant because such extreme hypotheticals are clearly unrealistic. I think that the mathematical idea of ...


"Hodge's general conjecture is false for trivial reasons."


The flattering lie You Could Have Invented Spectral Sequences by Timothy Y. Chow.


A minus sign that used to annoy me but now I know why it is there by Peter Tingley.


I find there is a world of difference between explaining things to a colleague, and explaining things to a close collaborator. With the latter, one really can communicate at the intuitive level, because one already has a reasonable idea of what the other person's mental model of the problem is. In some ways, I find that throwing out things to a ...


One that comes immediately to mind is Can one hear the shape of a drum?


Everyone I know in the AMS would like to make MR/MathSci free, but the problem is that it costs millions of dollars to produce and maintain (it requires a large staff in Ann Arbor and elsewhere, including many mathematicians), and no one has managed to find any other way to pay for it*. This is certainly something the mathematicians in the AMS are aware of ...


Mark Van Raamsdonk's Princeton PhD thesis in string theory was called "Making the most out of zero branes and a weak background". Priceless.


John Nash's "Equilibrium Points in n-Person Games" is only about a page and is one of the most important papers in game theory.


I've been spending a fair amount of time editing a journal this year, and it's pretty amazing what different people think of as a "referee report". The first thing you should keep in mind is that the editors will be incredibly appreciative if you look at the paper in detail and send in comments in a timely manner, whatever the comments are. In my mind ...


Final addition: Since I've produced many rambles, I thought I'd close my (anti-)contribution with a distilled version of the example I've attempted below. It's still something very standard, but, I hope, in the spirit of the original question. I'll describe it as if it were a personal thing. Almost always, I think of an integer as a function of the primes....



Here are some that came to mind: Equivalence. Basically, the idea that two things can be functionally equivalent (or close to equivalent) even if they look very different (and conversely, that two things can be superficially similar but functionally quite distinct). For instance, paying off a credit card at 10% is equivalent (as a first approximation, at ...


It's not a paper, and it's not groundbreaking, but it's short! A One-Sentence Proof That Every Prime $p\equiv 1\pmod 4$ Is a Sum of Two Squares D. Zagier The American Mathematical Monthly, Vol. 97, No. 2 (Feb., 1990), p. 144


John Stallings' "How not to prove the Poincare Conjecture" is lovely.


be self-critical. Recognize that constantly questioning one's own arguments, and those of authority figures, far from being corrosive or disrespectful, is the best way to strengthen those arguments.


I get this nominee from Halmos... E. Nelson, "A Proof of Liouville's Theorem", Proc. Amer. Math. Soc. 12 (1961) 995 9 lines long. Not the shortest paper ever, but maximizes importance/length ...


This is really just an extension of James D. Taylor's comment on the question, but recognizing the value of definitions, and the inherent ambiguity without them, is ridiculously helpful. Define your terms! I recently saw a talk on how to teach students to write mathematics well. The advice "think like a lawyer" was given, which I totally agree with when ...


Given the atmosphere of terror and fear in recent years, I did a double take when I first glanced at Bruce Berndt's paper "Ramanujan's association with radicals in India".


H. Lebesgue, Sur une généralisation de l’intégrale définie, Ac. Sci. C.R. 132 (1901), 1025– 1028. The beginning of measure theory as we know it, and a very short paper.


Paul Cohen's paper "The independence of the continuum hypothesis" in which he introduced forcing. Six pages long (and another six in the second paper, a year later) that completely changed logic and set theory. JSTOR access (may require a paywall) While I'm at it, two more in set theory: Kurt Goedel's proof of the consistency of the continuum hypothesis ...


I think your question is so important as to deserve multiple answers, even if there is a good deal of overlap among them. (Indeed, overlap indicates that various respondents feel the same way about something, which is important.) So: 1) Should you summarize the main results and or the argument? If so, how much is a good amount? What purpose does this ...


The first priority is to state your main results and explain why they are interesting (e.g. how they fit in with related work). With only 15 minutes you do not have much time to discuss proofs, but it is nice to give a brief outline of the proof of your main result and what is involved. As a first-time presenter, I would watch out for the following: First,...


And in the graph theory corner we have the famous Harary/Read paper "Is the null-graph a pointless concept?"


"On $O_n$" by D.E. Evans. ($\mathcal{O_n}$ is notation Cuntz gave for the algebras he introduced in "Simple $C^*$-algebras generated by isometries".)


I've refereed at least a dozen papers in my (short) career so far and I still find the process completely baffling. I'm wondering what is actually expected and what people tend to do... I used to think that I had to check a paper's correctness, but now I think that the main point of my report is to help the editors decide whether they should accept or ...

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