# Titles composed entirely of math symbols

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! :-)

• 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). Jul 11, 2011 at 5:55
• On "Fun with $\mathbb{F}_1$" it's worth noticing that the French for "1" is "un".. :) Jul 11, 2011 at 14:45
• 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. Jul 11, 2011 at 19:13
• 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. Mar 31, 2012 at 18:21
• Jeff: Cool! Let's collectively try to track it down... Mar 31, 2012 at 23:48

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

• @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--$SL_2(R)$." Jul 11, 2011 at 0:54

!

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

• Along this line, Doron Zeilberger gave a talk for an REU program about nothing, 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. Jul 11, 2011 at 11:47
• 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.") Jul 11, 2011 at 0:53
• I guess "the largest number not expressible as" offers several opportunities... Jul 11, 2011 at 1:07
• 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
– user9072
Jul 11, 2011 at 1:25
• This gets my third vote. Gerhard "Email Me About System Design" Paseman, 2011.07.11 Jul 11, 2011 at 17:48
• @Joseph: I think so, see oeis.org/A022566 . Dec 20, 2011 at 1:20
• "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." Jul 11, 2011 at 10:55

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.

$R(4,5)=25$

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).

• In the same vein, $R(5,5) \le 48$ was posted yesterday. Mar 28, 2017 at 13:33

Would IP=PSPACE count?

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

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

• 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. Sep 28, 2012 at 8:14
• There is Lang's "SL$_2(\bf R)$", which is the highest-voted answer to this question (by David Roberts). Mar 28, 2017 at 15:44

&

Yes, this is the title. Just "&". :-)

From Mulvey's homepage: "This paper, presented at the Topology Meeting in Taormina, Sicily in April, 1984, introduced the concept of quantale, outlining the programme of work in the spectral theory of C*-algebras and the constructive foundations of quantum mechanics to which it was expected to contribute. The paper is a slight development of that which appeared in the Tagungsbericht of the Category Meeting at Oberwolfach in September, 1983. It is included here since, although often quoted, it is more difficult to obtain in its published form in the Rendiconti del Circolo Matematico di Palermo. "

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

• That's a nice one. I heard of this once, but completely forgot. Nice to be reminded.
– user9072
Jul 11, 2011 at 1:33
• 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 $x (x+1) = \binom{n}{4}$." Jul 11, 2011 at 1:34
• Jul 11, 2011 at 9:54

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

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

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

http://arxiv.org/abs/1105.0825

This just in (https://arxiv.org/abs/1703.08768):

$$R(5,5) \leq 48$$

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

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

EDIT: As of 3 Oct 2016 "This paper has been withdrawn due to an error in the proof of Claim I.3.5"

"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 my favorite titles "$$A=B$$").

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

• +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.
– user9072
Jul 11, 2011 at 13:47

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.

$H_8$, 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.

Close but no cigar: On $O_n$, DE Evans RIMS, Kyoto Univ 16 (1980) 915-927, and its sequel On $𝑂_{𝑛+ 1}$, H Araki, AL Carey, DE Evans J. Operator Theory 12 (2) (1984), 247-26.

$O_n$ (Oh, not zero) is the Cuntz C*-algebra. I thought this was a very clever title at the time.

$\Gamma_4=0$

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.

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.

• Ha! I wonder how many other pithy paper titles were quashed by referees... 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. Dec 16, 2011 at 11:52
• Michael Rosen and I wanted to publish, with title “Ordinary Abelian varieties and $u$ and $I$”, but that was quashed, @JosephO'Rourke . Jan 17, 2020 at 18:10
• @Lubin: $u$ & $I$: Ha! Jan 17, 2020 at 18:18

$K_1(A,B,I)$

S. Geller, C. Weibel, J. Reine Angew. Math. 342 (1983), 12–34.

$K(A,B,I)$: II

S. Geller, C. Weibel, K-Theory 2 (1989), no. 6, 753-760.

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

$\mathcal{C}(A)$

PhD thesis about posets of commutative C*-subalgebras. According to some P.R. agent at the Radboud University, it might be the shortest title for a Dutch PhD thesis, but I am not completely sure whether or not that is true.

Not a paper, but Dan Freed's talk 4-3-2 8-7-6 about topological and conformal field theories.

• Neat. Naturally if we allow talk titles then there's much bigger scope for this kind of thing; I've given a few such talk titles myself, including "$6561101970383!$" (see mathoverflow.net/questions/19170 ). Mar 28, 2017 at 14:39

Jacques Roubaud has a book named $\in$, published by Gallimard in Paris 1967. It's not listed on his English wiki page (and is a pain to google if, like me, you've forgotten his name).

Can't add images here yet but here is a link to a pic of the cover: (French Amazon)

MIP*=RE

Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, Henry Yuen. arXiv Abstract. 13 Jan 2020.

...a negative answer to Tsirelson's problem... our results provide a refutation of Connes' embedding conjecture...

$K_{i}^{loc}(\mathbb{C})$, $i = 0, 1$, by Nicolae Teleman (link).

The monograph $Lx=b$ by Nisheeth Vishnoi (here), on fast ways to solve Laplacian systems.