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I apologize for burdening MO with such a vapid, nonresearch question, but I have been curious ever since Suvrit's popular October 2010 Most memorable titles MO question if there were any "$E=mc^2$-titles", as I think of them—how Einstein in retrospect might have entitled his 1905 paper (instead of "Zur Elektrodynamik bewegter Körper"!)—paper/book titles composed entirely of math symbols.

There are two close misses in the responses to that MO question: Connes et al.'s "Fun with $\mathbb{F}_{1}$", and Taubes's "${\rm GR}={\rm SW}$: Counting curves and connections." The only title entirely composed of math symbols with which I'm familiar is the delightful book A=B, by Marko Petkovsek, Herbert Wilf, and Doron Zeilberger. Can you identify others?

Please interpret this question in a weekend-recreational spirit! :-)

edited Jul 11 2011 at 1:45

Joseph O'Rourke

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If Chaitin came out with a book called $\Omega$, that would be the last word in "$E=mc^2$-titles (sorry, couldn't help myself). – David Roberts Jul 11 2011 at 5:55

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On "Fun with $\mathbb{F}_1$" it's worth noticing that the French for "1" is "un".. :) – domenico fiorenza Jul 11 2011 at 14:45

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The "most memorable titles MO question" was only secondarily a request for examples of titles, but that secondary question was the only one that people answered, until after a large number of such answers had appeared. I think I posted at least two such examples that were favorable viewed, but then I posted something that was closer to the primary thrust of the question. I was severely and in fact abusively taken to task for not staying on topic, by someone who would have known that I was in fact on topic if he had read the question. – Michael Hardy Jul 11 2011 at 19:13

.....and the design of the software, and to some extent the culture, of MO, tacitly encourages such abuses. – Michael Hardy Jul 11 2011 at 19:13

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A few months ago, I ran across a preprint whose title consisted two simple figures separated by an equals sign. Of course, now I've forgotten the authors. – JeffE Mar 31 2012 at 18:21

show 2 more comments

David Roberts · Answer 1 · 2011-07-11 00:27:19Z UTC

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$SL_2(\mathbf{R})$ (link)

answered Jul 11 2011 at 0:27

David Roberts

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@David: Great example! "$SL_2(R)$ gives the student an introduction to the infinite dimensional representation theory of semisimple Lie groups by concentrating on one example--$SL_2(R)$." – Joseph O'Rourke Jul 11 2011 at 0:54

quid · Answer 2 · 2011-07-11 00:29:07Z UTC

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7 373 170 279 850

answered Jul 11 2011 at 0:29

quid

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Ha! Has their conjecture stood fast these past dozen years? (Abstract: "We conjecture that 7,373,170,279,850 is the largest integer which cannot be expressed as the sum of four nonnegative integral cubes.") – Joseph O'Rourke Jul 11 2011 at 0:53

I think so. At the moment it is difficult for me to search literature. But in a similar spirit, it is known that 13792 is the largest number not expressible as a sum of 16 biquadrates; Deshouillers, Kawada, Wooley, Mém. Soc. Math. France 100, 2005. – quid Jul 11 2011 at 0:59

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I guess "the largest number not expressible as" offers several opportunities... – Joseph O'Rourke Jul 11 2011 at 1:07

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Let me explain why this precise problems are considered: By a classical result of Hilbert (solving Waring's problem) every nonnegative integer is a sum of a fixed number of k-th nonnegative integral powers. (The fixed depends of course on the k). One can now ask what is the best 'fixed' for a given k. It turns out that small numbers cause most problems and one gets by (for given k) with a smaller number of k-th powers, if one just wants all sufficiently large integers as a sum. Now, this raises the question, what is the 'sufficiently large'. See en.wikipedia.org/wiki/Waring's_problem – quid Jul 11 2011 at 1:25

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@Joseph: I think so, see oeis.org/A022566 . – Charles Dec 20 2011 at 1:20

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Chandan Singh Dalawat · Answer 3 · 2011-07-11 11:20:21Z UTC

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!

(Title of a talk about the factorial function by Manjul Bhargava at the Clay conference in Paris in the year 2000.)

answered Jul 11 2011 at 11:20

Chandan Singh Dalawat

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Along this line, Doron Zeilberger gave a talk [for an REU program about nothing](math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/…), and his title was, well, nothing at all. (Not "nothing". But nothing.) Unfortunately, I am not sure if for this exercise of Joseph's, whether "consisting of mathematical symbols" requires the subset to be non-trivial. – Willie Wong Jul 11 2011 at 11:47

J. H. S. · Answer 4 · 2012-09-28 02:38:42Z UTC

Professor Luca and his co-authors are surely fond of this kind of titles:

F. Luca & B. de Weger, $\sigma_k(F_m)=F_n$. New Zealand J. Math. 40 (2010), 1–13.
F. Luca & F. Nicolae, $\phi(F_n)=F_m$. Integers 9 (2009), A30, 375–400.
F. Luca & M. Mignotte, $\phi(F_{11})=88$. Divulg. Mat. 14 (2006), no. 2, 101–106.
F. Luca & P. Stănică, $F_1F_2F_3F_4F_5F_6F_8F_{10}F_{12}=11!$. Port. Math. (N.S.) 63 (2006), no. 3, 251–260.

Dirk · Answer 5 · 2011-07-11 07:19:56Z UTC

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$H=W$

answered Jul 11 2011 at 7:19

Dirk

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"Two concepts which are often used in the theory of partial differential equations and the calculus of variations are the so-called $H$ spaces and $W$ spaces." – Joseph O'Rourke Jul 11 2011 at 10:55

Richard Kent · Answer 6 · 2011-07-11 05:22:37Z UTC

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McCarthy, Charles A. $c_p.$ Israel J. Math. 5 1967 249–271.

answered Jul 11 2011 at 5:22

Richard Kent

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MathSciNet link: ams.org/mathscinet-getitem?mr=MR0225140 – Joseph O'Rourke Jul 11 2011 at 10:57

mhum · Answer 7 · 2011-07-11 07:23:39Z UTC

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Would IP=PSPACE count?

answered Jul 11 2011 at 7:23

mhum

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Jonny Evans · Answer 8 · 2011-07-11 06:27:51Z UTC

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$\int_x^{hx}(g^*\alpha-\alpha)$ (by Kedra and Gal)

http://arxiv.org/abs/1105.0825

answered Jul 11 2011 at 6:27

Jonny Evans

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Brendan McKay · Answer 9 · 2011-12-16 07:33:42Z UTC

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$R(4,5)=25$

B. D. McKay and S. P. Radziszowski, J. Graph Theory, 19 (1995) 309-322.

The title is also the main theorem. $R(4,5)$ is a classical Ramsey number (the one most recently determined exactly).

answered Dec 16 2011 at 7:33

Brendan McKay

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Gerhard Paseman · Answer 10 · 2011-07-11 01:20:17Z UTC

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210=14*15=5*6*7

I may have the title wrong. It is about the simultaneous solution of some Pell-like equations. I will provide more detail as my memory permits.

Gerhard "Email Me About System Design" Paseman, 2011.07.10

answered Jul 11 2011 at 1:20

Gerhard Paseman

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That's a nice one. I heard of this once, but completely forgot. Nice to be reminded. – quid Jul 11 2011 at 1:33

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Pinter and deWeger, "$210 = 14 \times 15 = 5 \times 6 \times 7 = \binom{21}{4} = \binom{10}{4}$". Publ. Math. Debrecen. 51(1-2) 175-189 (1997). "It is given all the solutions to the Diophantine equations $(y−1)y(y+1)=\binom{n}{4}$ and $x (x+1) = \binom{n}{4}$." – Joseph O'Rourke Jul 11 2011 at 1:34

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From mathoverflow.net/questions/44326 : deweger.xs4all.nl/papers/… – Goldstern Jul 11 2011 at 9:54

monodromy · Answer 11 · 2011-12-19 16:42:04Z UTC

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$H_g^1(K,V)=H_{st}^1(K,V)$

An unpublished manuscript by Osamu Hyodo (who passed away untimely).

answered Dec 19 2011 at 16:42

monodromy

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Chandan Singh Dalawat · Answer 12 · 2011-12-16 03:30:25Z UTC

$H_8$, by Jacques Martinet.

$GL_n$, by William Casselman.

Both these articles appear in the a book edited by Albrecht Fröhlich: Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), pp. 525–538. Academic Press, London, 1977.

David Roberts · Answer 13 · 2012-03-15 00:26:28Z UTC

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$$\left(1+\frac{d}{dz}\right)^{-1}$$

only a preprint, though: http://arxiv.org/abs/1203.3045

answered Mar 15 2012 at 0:26

David Roberts

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Peter Arndt · Answer 14 · 2012-03-31 01:59:16Z UTC

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"Pi" (I keep "A source book" in parentheses to hide the non-mathematical part), L. B. Berggren, J. M. Borwein, P. B. Borwein (Eds.).

"Z=60", Conference in Honor of Doron Zeilberger's 60th Birthday (this, of course, is influenced by one of ma favorite titles "$A=B$").

Removed (following the healthy criticism): "2012", a 2009 American science fiction disaster movie.

edited Mar 31 2012 at 1:59

Peter Arndt

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3 revisions, 2 users
Wadim Zudilin 69%

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+1 for Pi and Zeilberger. And, if works of fiction count, I think one should add 2001 (and 2010, 2061, 3001); let's ignore the odyssey add-ons. – quid Jul 11 2011 at 13:47

Point taken! Thanks – Wadim Zudilin Jul 11 2011 at 13:51

François Brunault · Answer 15 · 2011-12-16 13:26:59Z UTC

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$\Delta=b^2-4ac$, by Jean-Pierre Serre (Math. Medley, Singapore Math. Soc. 13, 1985, 1-10).

answered Dec 16 2011 at 13:26

François Brunault

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J.-P. Serre also has a book entitled `$SL_2$'. No wait, come to think of it the title also mentions `Arbres, amalgames', whatever they are. – shane.orourke Sep 28 at 8:14

Aaron Mazel-Gee · Answer 16 · 2012-01-17 09:36:24Z UTC

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Here is $H_\infty\not= E_\infty$, wherein Justin Noel gives an example of an $H_\infty$-structure on a ring spectrum which does not descend from an $E_\infty$-structure.

answered Jan 17 2012 at 9:36

Aaron Mazel-Gee

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Alain Valette · Answer 17 · 2011-07-11 15:48:47Z UTC

2

$\Gamma_4=0$

is the subtitle of Jean Cerf's famous lecture notes: Sur les difféomorphismes de la sphère de dimension trois $(\Gamma _{4}=0)$. (French) Lecture Notes in Mathematics, No. 53 Springer-Verlag, Berlin-New York 1968 xii+133 pp.

answered Jul 11 2011 at 15:48

Alain Valette

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Emil Jeřábek · Answer 18 · 2011-10-20 17:58:31Z UTC

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I apologize for a bit of vanity, which, worse yet, is not even a proper example: I nearly published a paper entitled $T^0_2(MSP)=PV_1$, but a referee made me rename it in the final version.

answered Oct 20 2011 at 17:58

Emil Jeřábek

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2

Ha! I wonder how many other pithy paper titles were quashed by referees... – Joseph O'Rourke Oct 20 2011 at 19:03

I wanted to publish a paper called "B-pairs and (φ,Γ)-modules" but the editors made me change it on the ground that they did not want too many math symbols in a title. – Laurent Berger Dec 16 2011 at 11:52

Nick Gill · Answer 19 · 2012-09-28 09:08:23Z UTC

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Thomas Forster's Phd thesis is called ``N.F.'' On his website he claims that this is the shortest title for a Cambrige maths PhD on record. The abstract is also pretty short.

answered Sep 28 at 9:08

Nick Gill

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Titles composed entirely of math symbols

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