# Which math paper maximizes the ratio (importance)/(length)?

My vote would be Milnor's 7-page paper "On manifolds homeomorphic to the 7-sphere", in Vol. 64 of Annals of Math. For those who have not read it, he explicitly constructs smooth 7-manifolds which are homeomorphic but not diffeomorphic to the standard 7-sphere.

What do you think?

Note: If you have a contribution, then (by definition) it will be a paper worth reading so please do give a journal reference or hyperlink!

Edit: To echo Richard's comment, the emphasis here is really on short papers. However I don't want to give an arbitrary numerical bound, so just use good judgement...

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You should probably bound the length, cuz otherwise you could just pick your favorite paper of Ratner, Grothendieck, Thurston, et cetera and the importance blows everything else away. – Richard Kent Dec 1 '09 at 1:41
Or Gromov, "from whose sentences people have written theses" (as I have seen someone write somewhere) – Mariano Suárez-Alvarez Dec 1 '09 at 2:31
The award for the corresponding question for paper titles would have to go to "H = W". Meyers and Serrin, Proc. Nat. Acad, Sci. USA 51 (1964), 1055-6. – John D. Cook Jan 6 '10 at 2:49
It also depends on what you define a "paper". A number of fundamental results have been announced, and their proof has been sketched, in the C.R. Acad. Sci. - and all of them are four pages long. – Delio Mugnolo Nov 9 '13 at 14:48
Golod, E.S; Shafarevich, I.R. (1964), "On the class field tower", Izv. Akad. Nauk SSSR 28: 261–272 – TT_ Sep 27 '14 at 20:14

## 68 Answers

The 1949 paper by R.C. Bose "A Note on Fisher's Inequality for Balanced Incomplete Block Designs" arguably gave birth to the linear algebra method in combinatorics which has since been used by many to solve highly non-trivial problems as discussed here: Linear Algebra Proofs in Combinatorics?

The paper is 2 page long: http://projecteuclid.org/download/pdf_1/euclid.aoms/1177729958

Here's a description of Bose and his work from the manuscript Linear Algebra Methods in Combinatorics by Babai and Frankl:

The affiliation listed on Bose’s paper is the Institute of Statistics, University of North Carolina. Before taking up residence in the U.S. in 1948, Bose worked at the Indian Statistical Institute in Calcutta. One of the most influential combinatorialists of the decades to come, Bose was forced to become a statistician by the lack of employment chances in mathematics in his native country. A pure mathematician hardly in disguise, he reared generations of combinatorialists. His students at Chapel Hill included D. K. Ray-Chaudhuri, a name that together with his student R. M. Wilson (so, may be a grandson of Bose?) will appear several dozen times on these pages for their far reaching extension of Bose’s method.

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What about Selberg's 1947 paper?

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A favourite of mine is Kobayashi and Wu's 4 page Annals paper "On holomorphic sections of certain Hermitian vector bundles" where they introduce the Bochner-method, which is nowadays used everywhere in differential geometry as an easy and effective method of proving vanishing results.

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Bochner method was introduced by Bochner in 1946-48 and used by numerous authors to prove vanishing theorems for cohomology groups of vector bundles ever since. This remark, of course, is not meant to diminish Kobayashi-Wu's 1970 paper. – Misha May 25 '13 at 18:37

I will vote for C.Fefferman's paper "The multiplier problem for the ball" http://mate.dm.uba.ar/~hafg/inter-u-2010/fefferman.pdf, which is only about 5 pages and he solved an open problem about multipliers, and he wrote this when he was only a teenager!

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Here is my list (in no specific order):

(*) A proof of Ehrenfeucht's conjecture about infinite systems of equations in free groups and semigroups by Victor Guba:
V.S.Guba "Equivalence of infinite systems of equations in free groups and semigroups to finite subsystems", Mathematical notes of the Academy of Sciences of the USSR, September 1986, Volume 40, 3, pp 688-690.

(*) A.A.Razborov, “Lower bounds on monotone complexity of the logical permanent”, Math. Notes USSR, 37:6 (1985), 485–493.
As Laszlo Lovasz put it in his talk "The Work of A.A.Razborov" (can be easily found on the Internet):
In an area where any step forward seemed almost hopeless (but which was at the same time a central area of theoretical computer science) his results meant that deep methods could be developed and to obtain strong lower bounds for algorithms was not impossible.

(*) Isaac Newton "The mathematical principles of natural philosophy" - in this case the (finite) length of the work does not matter, since the importance is infinite :)

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Maybe the paper of R. Brauer and Fowler, K. A. (1955): "On groups of even order", Annals of Mathematic, Second Series 62: 565–583, ISSN 0003-486X, JSTOR 1970080, MR 0074414 deserves a mention since this is generally accepted as the point when it was realised the Classification of the Finite Simple Groups might be a feasible project.

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I suppose the word "importance" in the equation can allow for some subjective input (some papers might be important to certain people, while to others not so important for their work).

This paper, entitled Finiteness of the number of compatibly-split subvarieties by Kumar and Mehta, is only 3 pages long:

http://arxiv.org/abs/0901.2098

For those who work with Frobenius splittings, it is an important result, one which was actually believed to be true for decades but not proven until 2009!

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How about Galois's letter written on the eve of his death and published by Liouville 17 years later?

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I think we should draw the line somewhere when it comes to what constitutes a paper. Otherwise we will have Archimedes's sketch on a bit of slate while he was down the taverna with the lads... – Yemon Choi Oct 2 '14 at 17:04
@Yemon: Or when he was explaining some Roman soldiers about circles on the beach... – Asaf Karagila Oct 2 '14 at 17:31
+1 Maybe we should draw a line, but how can we not to mention the Galois's letter??? – TT_ Oct 7 '14 at 20:05