MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Different people like different things in math, but sometimes you stand in awe before a beautiful and simple, but not universally known, result that you want to share with any of your colleagues.

Do you have such an example?

Let's try to go in the direction of papers that can actually be read online or accessible with little effort, e.g. in major libraries, so that people could actually follow your advice and read about it immediately.

And as usual let's do one per post and vote freely, vote a lot.

share|cite|improve this question

closed as off topic by Kevin H. Lin, Andrés E. Caicedo, Felipe Voloch, Hailong Dao, Bill Johnson Jul 9 '11 at 15:01

Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question.

Why are so many answers big-picture papers and philosophical tracts? I'm sure many of them are good papers, but is this really what the question was about? Am I right in suspecting that posters only read the title of the question and not the question itself? – Thierry Zell Sep 4 '10 at 0:23
Perhaps it's time to close this question. – S. Carnahan Oct 22 '10 at 17:40
Agreed, as Thierry and Tobias say, there are too many recommendations for punditry. – Robin Chapman Nov 17 '10 at 11:48

41 Answers 41

Cannon's beautiful and accessible paper "The combinatorial structure of cocompact discrete hyperbolic groups" was one of the original impetuses for geometric group theory. It inspired many people (including me) to become interested in infinite discrete groups. It is available here:

share|cite|improve this answer

2N Noncollinear Points Determine at Least 2N Directions, by Peter Ungar. This is a beautiful short paper that proves the result in the title.

A general remark: If you have to choose a single paper (or a single paper of a mathematician selected in other answers), I would recommend more strongy to choose original papers of important basic results rather than large survey papers or "meta" paper about mathematics. (This is also closer to the original intention of the question.)

share|cite|improve this answer
Indeed, you get correctly the original intent, though I think some reviews are original enough in their scope and composition to be considered essentially primary sources for some problems. – Ilya Nikokoshev Nov 30 '09 at 15:09

The Unreasonable Effectiveness of Mathematics in the Natural Sciences

by Eugene Wigner

Although Wigner is physicist, I consider this article about mathematical physics very important both for physicists and mathematicians. It's a wonderful feeling to realize to what extent our world can be mathematical.

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning. E.Wigner

share|cite|improve this answer
This is not only a historically important paper,it focuses on an aspect of mathematics that Western culture has hesitated to return to after the wholesale rejection of it during the Bourbaki era. – The Mathemagician Oct 22 '10 at 19:04
Even if bashing Bourbaki seems to have become hip again (as it was actually during the Bourbaki era as well), and thus the myth of the "wholesale rejection" of applications of mathematics "during the Bourbaki era" has become generally accepted in certain circles, it still remains a myth, which like all myths contains a germ of truth surrounded by a lot of prejudices, misunderstandings and plainly wrong statements. – Tobias Hartnick Nov 17 '10 at 13:44
Ah, I can't resist pointing to a personal favorite counter-article to Wigner's: "How effective indeed is present-day mathematics?" It would certainly be in accordance to the OP. I read it back then by chance, out of misdirected curiosity (just because a teacher of mine wrote it), but I think it was what paved the road for me to go on realizing that a kuhnian view on mathematics is not necessarily a preposterous or a blasphemous idea. – Basil Jul 2 '14 at 22:01

I highly recommend this lucid, little book (with the length of a paper):
Mathematics: A very short introduction, by Fields Medalist Timothy Gowers

share|cite|improve this answer

"An Introduction to the Conjugate Gradient Method Without the Agonizing Pain" by Jonathan Shewchuk at UC Berkeley

share|cite|improve this answer

Carl's Pomerance "A tale of two sieves", available at

It makes a quick introduction to subexponential factoring algorithms via their development from Fermat's Algorithm and then compares the Quadratic Sieve with Her Majesty the (General) Number Field Sieve, in a thorough, appealing and very understanable manner.

share|cite|improve this answer

PDE as a Unified Subject by Sergiu Klainerman.
An essay on partial differential equations written by a leading expert in the field, I strongly recommend to anyone who aspires to know more on the subject as well as to those who are not interested strictly in PDE's, but would like to get a grasp of interactions between Mathematics and Physics. There are also many interesting references.

share|cite|improve this answer

Toen's course on stacks. I don't know if this counts as a paper, but courses 2,3, and 4 introduce a really interesting approach to geometry using the functor of points approach that I've not seen before.

share|cite|improve this answer
Your link is a 404... – Julien Puydt Jul 5 '11 at 20:17
After some searching, I could lay my hand on it : – Julien Puydt Jul 5 '11 at 20:19

That's easy just off the top of my head,Illya: Nets And Filters In Topology by the late Robert G. Bartle;appearing in the 1955 Volume 62 of American Mathematical Monthly. I remember having a friend in the Stanford mathematics honor society who'd published papers by age 20,but had never heard of either nets or filters. I recommended it to him right on the spot.

share|cite|improve this answer
As came up in a different question recently, this paper has some missteps. (In fact it has an erratum, albeit published 8 years later.) I agree that it is worth reading, but I would recommend also Smiley's Filters and equivalent nets, American Math. Monthly, 1957. – Pete L. Clark Jul 14 '10 at 20:24

On the theorem of Pythagoras by by E.W. Dijkstra. (Did you know that in every plane triangle sgn$(\alpha + \beta - \gamma)$ = sgn$(a^2 + b^2 - c^2)$, a "theorem, say, 4 times as rich [as the original]"?)

share|cite|improve this answer
I don't think this is any news to most mathematicians. This is even in some good German schoolbooks from the 1960's-70's. Often when there is a statement like "if $a=b$ then $c=d$" one could check whether $a\leq b$ implies $c\leq d$, and lots of geometric inequalities have been created this way from identities. – darij grinberg Nov 17 '10 at 14:08
Doesn't this "richer" statement follow easily from the law of cosines? – Federico Poloni Jul 6 '11 at 10:23

"Rigor and Proof in Mathematics: A Historical Perspective" by Israel Kleiner. Mathematics Magazine December 1991, 64:291-314.

This paper gives a very nice overview of how the understanding of rigor in mathematics has evolved from the early ages to the 20th century.

share|cite|improve this answer

Not the answer you're looking for? Browse other questions tagged or ask your own question.