## Hot answers tagged math-communication

126

A natural choice is Riemann's "On the Number of Primes Less Than a Given Magnitude" at only 8 pages long...
http://en.wikipedia.org/wiki/On_the_Number_of_Primes_Less_Than_a_Given_Magnitude

90

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 ...

75

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.

73

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 ...

59

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
http://www.jstor.org/pss/2323918

56

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 ...
http://www.jstor.org/stable/2034412

52

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.

51

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 ...

45

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 ...

43

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 ...

37

Riemann's Habilitationsschrift, On the hypotheses which lie at the foundation of geometry, was the start of Riemannian Geometry. An English translation took up 6 pages in Nature.

37

Whenever someone claims a proof (or disproof) of a big conjecture, many people leap to the question of whether the proof is correct. The problem then is it that it takes an enormous amount of work to confirm that a proof is correct. Even a clear mistake in a proof could be reparable. Moreover, attempted proofs have inferences that amount to gaps of ...

36

try this, the latest in a long line of recreational mathematics on the topic
"Keeping Dry: The Mathematics of Running in the Rain"
Dan Kalman and Bruce Torrence,
Mathematics Magazine, Volume 82, Number 4 (2009) 266-277

35

This was discussed a little on the algebraic topology list last autumn, you can look up the archives to see what was said.
Technologically, this is easy. The problems come in when you think beyond that.
How would such a site start? Initially, there would be very few reviews so no one would have a reason to visit the site (the probability of the paper ...

34

The fact that giving a presentation at the black board slows down the speed of presentation and helps the audience to digest the stuff is related to the fact that mathematicians use a language that has a high information density (formulas, diagrams) compared to other fields. Moreover the symbols and other structures used in that language CAN be written on a ...

32

For me it depends on the type of talk. But if I'm giving either a lecture or a seminar that involves presenting proofs (or sketches of proofs) then I like blackboards best. There are many reasons for this, but three important ones are (i) it forces me to understand the material well enough to memorize it (I don't use notes for talks, and what I say applies ...

31

Noam Elkies, The existence of infinitely many supersingular primes for every elliptic curve over Q, Invent. Math. 89 (1987), 561-568.

30

Depending on how strict you are, this might not qualify as a paper. Hilbert's 1900 ICM talk in which he posed his 23 problems.

30

One of the shortest papers ever published is probably John Milnor's Eigenvalues of the Laplace Operator on Certain Manifolds, Proceedings of the National Academy of Sciences of USA, 1964, p. 542
He shows that a compact Riemannian manifold is not characterized by the eigenvalues of its Laplacian. It takes him little more than half of a page.

30

I often give this to Freshmen. Learning the right way to start with is easier than trying to change later.

30

1) Should you summarize the main results and or the argument?
In general, I would say: No.
However, for some journals I have come to know the overwhelmed editors who in fact need
this summary. So: Write it for the editors, if you write it at all.
[But see my acquiescence to Jukka Suomela's compelling point in the comments.]
2) What do you do when ...

30

There are only 2 rules that really matter when refereeing:
(1) Either referee it within a couple of weeks, or reply immediately saying you do not have time.
(2) Assume that the author will see your report and will find out who wrote it.

29

As others have indicated, the only 100% effective method of preventing getting "scooped" or finding out that your result already exists in the literature is that of complete abstinence: i.e., not trying to do any research at all.
Obviously this method is far too draconian for most of us on this site. I want to support statements of Gowers and Nielsen: ...

28

Endre Szemeredi's paper on the Regularity Lemma is just 3 pages long. I think that is a good candidate as well.
Szemerédi, Endre (1978), "Regular partitions of graphs", Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), Colloq. Internat. CNRS, 260, Paris: CNRS, pp. 399–401,

28

Kahn and Kalai's, "A counterexample to Borsuk's conjecture" is a 3-page paper which settles a sixty-year-old conjecture with an explicit counterexample in $\mathbb{R}^{1325}$ (and in all sufficiently high dimensions). Although the paper is 3 pages, most of that is background on the problem and references --- the construction itself is only one paragraph.
...

27

I've been using Dropbox for a while. It works like this: You sign up at their webpage for free to get 2gb space which you can access via their webinterface. If you want (and this is the novel part) you can install their software, which sets up a folder on your computer which is automatically (and in the background) synchronized with their server. You can ...

26

Kazhdan's paper "On the connection of the dual space of a group with the structure of its closed subgroups" introduced property (T) and proved many of its standard properties. And it's only 3 pages long (and it contains a surprisingly large number of details for such a short paper!)

25

The one-page paper
Golay, Marcel J. E.: "Notes on Digital Coding", Proc. IRE 37, p. 657, 1949,
which introduces the Golay code.

25

John Stillwell's recent book Naive Lie Theory is amazing and in a similar vein. It provides great geometrical intuition for many of the common matrix groups. What is particularly impressive about this book is how he motivates more complicated ideas, such as maximal torii in a very elementary fashion. It is perfect for undergrads looking for a good ...

24

Rather than creating a "free alternative for MathSciNet", it seems more useful to do what MathSciNet does not, e.g. make a site that maintains lists of errors in published papers.
I personally think MathSciNet is a great product, very much worthy of support, and I do not get the argument "since MathSciNet is not freely available, one should not support ...

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