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1.

Recently I was asked what the shortest mathematics Ph.D. thesis on record is. This is not the first time I have encountered this question—it seems to be perennially fascinating to research mathematicians—but as you can imagine, finding the answer to such an urban-legendy question is not easy. Googling around produces a lot of dubious leads that are sometimes difficult to confirm or disconfirm, since Ph.D. theses are often not published.

MO seems like a good place to answer such a question definitively. It is similar in genre to a question recently asked by Greg Kuperberg that tries to straighten out the facts about a particular widely circulated urban legend.

However, after seeing several discussions here on meta about non-technical questions, I get the impression that a sizable number of regular participants don't want questions like this on MO. Should I pose it or not?

2.
One recent Harvard Ph.D. thesis is 36 pages long.

By the way, are there any serious contenders for longest thesis besides Kai-wen Lan's (which clocks in at a face-melting 1027 pages)?
3.
Whoops, Eva Kallin's thesis is 14 pages. Maybe this was one of your dubious leads...
4.
My preference is for this question not to appear on MO. Advertising a short dissertation could appear like a bit of a publicity stunt. Moreover, this appears like a slippery slope: eventually we'll be comparing how many papers different mathematicans have in the Annals, how many grad students people have had, how many of them got jobs at Harvard, Oxford, Stanford, etc...

Moreover, although a short dissertation is kind of a novelty, it's not really all that interesting. The Fary-Milnor theorem is an interesting bit of mathematics and its history is significant. A short dissertation is not.
5.
On the question of how short a math dissertation could, theoretically, be, we have Littlewood, pages 39-40 of the Miscellany:

The question recently arose in conversation whether a dissertation of 2 lines could deserve and get a Fellowship. I had answered this for myself long before; in mathematics the answer is yes.

Cayley's projective definition of length is a clear case if we may interpret "2 lines" with reasonable latitude. With Picard's Theorem it could be literally 2, one of statement, one of proof.

(Theorem.) An integral function never 0 or 1 is constant.

(Proof.) exp{i Omega(f(z))} is a bounded integral function.

(tau = Omega(w) inverse to w = k^2(tau)).

The last bracket is needed solely because of the trivial accident that the function Omega, unlike its inverse k^2(tau), happens to have no unmistakable name.

Littlewood goes on to explain the k^2(tau) is the modular function etc., etc.
6.

@Ryan: Questions like that are unlikely to appear on MO because they are readily answerable via the Mathematics Genealogy Project or MathSciNet. The short Ph.D. thesis question is not so easy to answer reliably.

I'm also not sure that I agree with you that a short dissertation is not interesting. It's true that it isn't necessarily interesting. However, provided that the dissertation was produced in good faith (as opposed to being a publicity stunt or a mere formality or something), it is about as interesting as a "short proof" is. And don't most of us feel that short proofs are interesting? For example, the short papers listed here (search for "Nelson" to get to the list) are all pretty interesting in my mind. (By the way, note that the "shortest paper" question doesn't have the same urban-legend tendencies because it's much easier to verify the facts.)

But let me state explicitly my main reason for wanting to ask this question on MO: I'm kind of sick of hearing this question asked yet again and having no rebuttal to urban-legendy responses of the form, "So-and-so's dissertation was only epsilon pages long!" where epsilon is a positive real much less than 1. It would be nice to put this question to rest.

Having said all that, I'll refrain from posing the question if a couple other people say they don't like it.

7.
I'm not terribly opinionated on this. But I am skeptical of the topic and figured I'd express a clear opinion rather that give you a lot of the dithering, internal dialogue in my brain...
8.
I like it more than most soft questions. I like that it has the potential of a definite answer.
9.
I'm also not convinced that the possibility of an interesting answer is enough to make a question suitable for MO.

For example, if the answer is a 1-line dissertation from the University of New Sarepta, Alberta, what would we have gained from that?
• CommentAuthorvoloch
• CommentTimeFeb 7th 2011

Here is a strong contender (info from MathSciNet, but I had heard of it before):

MR2615548
Martens, Henrik Herman Buvik
A NEW PROOF OF TORELLI'S THEOREM.
Thesis (Ph.D.)–New York University. 1962. 12 pp.

The problem with the question is that we can always give examples but, unless someone presents us with an empty thesis, we can't guarantee it is an absolute minimum. Maybe someone with strong mathscinet-webscraping-fu (how about that for a neologism!) can find the minimum among the theses listed there.
10.
I can't seem to find the actual thesis version, but MR0903384 (Elkies, Noam D. "The existence of infinitely many supersingular primes for every elliptic curve over Q." Invent. Math. 89 (1987), no. 3, 561–567.) is only 7 journal pages.
• CommentAuthorvoloch
• CommentTimeFeb 7th 2011

I guess Elkies was a bit more verbose when he wrote his thesis:

SUPERSINGULAR PRIMES OF A GIVEN ELLIPTIC CURVE OVER A NUMBER FIELD
by ELKIES, NOAM DAVID Ph.D., Harvard University, 1987, 41 pages; AAT 8800772

Info from: http://disexpress.umi.com/dxweb

The question of what is the shortest PhD thesis is different from the question of what is the shortest paper originating from a PhD thesis. I think I could (but won't) find examples where the latter is the empty set.
11.
@Voloch: Taylor never published his thesis.
12.

I think the question is fine. It is soft, but it is closely linked to the culture of mathematics, and I think it will be of interest to many.

(It is certainly at least as good as a previous question of mine, "Which pair of mathematicians has the most joint papers?")

• CommentAuthorvoloch
• CommentTimeFeb 7th 2011

@David. Taylor is a very common surname. I could perhaps find an example with that surname. If you meant Richard Taylor, I don't think that's the kind of example I had in mind.
13.
@Voloch: I did indeed mean Richard Taylor. Ahh, I believe I mistook your meaning: were you suggesting the existence of theses free of any publishable results?
14.

@Voloch: When I look at Martens' thesis (it is freely available online), it seems longer than 12 pages to me... 32 leaves seems to be about right to me based on the scanned pdf I downloaded (this includes introduction, appendix, bibliography, etc). I downloaded it from http://www.archive.org/details/newproofoftorell00mart

Regarding the OP's query, I find this question to be very interesting. Having said that, I generally prefer fewer such questions (i.e., nontechnical) on MO. So I will sit on the fence.

• CommentAuthorvoloch
• CommentTimeFeb 8th 2011

@Philip: Thanks for the link. I downloaded the document you found and I am puzzled. It's possible that this is not the thesis but a longer document. It's an Air Force grant research report. I am not going to buy the ProQuest version of the thesis, linked from MathSciNet just to figure out what's going on. I had heard the story that Martens had written an unusually short thesis (don't recall if an actual page count was mentioned) from someone who was in NY at the time and knew Martens. I don't know.
15.
I would like to know the answer to this question.
16.

@Voloch: I think that you are right about the document that I linked not being the thesis after all. Besides the fact that it actually says that it is a report (which I obviously didn't look at closely enough the first time!), it is lacking other things that one might expect to find in a dissertation (acknowledgements, declaration of originality, etc).

I apologize for any confusion brought about by my earlier claim that it was Martens' thesis.

17.

I'd always heard that David Rector's thesis (at MIT) was remarkably short, but I can't seem to access it from the web. The publication based on it (Rector, David L. An unstable Adams spectral sequence. Topology 5 1966 343–346) could only be called 4 pages if one was generous.

18.
IMO the thread should probably have the soft-question tag, as it's not an actual mathematics question.
• CommentAuthorlandsburg
• CommentTimeFeb 8th 2011

I have heard that the substantive portion of Pete Tomas's thesis at Princeton (circa 1979 or so) was shorter than a (spectacularly good) page, but I'm not sure how "substantive portion" was to be interpreted here.
19.
John Nash comes in at 27
20.
*Sigh*... How did this META discussion end up being a discussion of the question itself?

Anyway, my rationale for saying that this question is appropriate for MO is that it's part of mathematical culture.
Mathematics dissertations tend to run short even on average, and I have trouble coming up with other disciplines in which
such low outliers might even happen. If you've ever had to deal with academics in other fields, you may end up having
to explain why this is so; not an easy task.

Having a specific reference to a record of conciseness can help there, and, as Timothy Chow rightly points out,
we've all heard stories, but verifiable claims are hard to come by.
21.
I had no idea that a ~30 page thesis is considered short enough to be noteworthy. I was under the (perhaps mistaken) impression that a decent research paper makes a decent thesis, and many good research papers are much shorter than 30 pages.

For what it's worth, my thesis was 29 pages (according to http://disexpress.umi.com/dxweb ; I don't have a copy of it). The published version was either 13 pages or 131 pages, depending on how you count it. The official thesis corresponds to Chapter 5 (13 pages) of the monograph (131 pages) that was eventually published. Since that chapter was largely independent of the rest of the work, I thought it would be less hassle to just turn that section in as my thesis.
22.

@David: Regarding your question about the longest thesis, wasn't Jacob Lurie's somewhere in the neighborhood of 1000?

23.

To bring the conversation back to the original purpose: I don't think that this is a suitable MO question. The Milnor legend is not a suitable case to base an argument on, and even if that was considered passable (I didn't see it until the matter had actually been resolved so please don't read anything into my not voting to close) then it was barely so. This question is considerably looser, considerably more pointless, and considerably harder to verify an answer to.

If this question appears on MO, I will vote to close.

24.

@Andrew Stacey: the question has already appeared on MO.

25.

... as has my vote to close.

(I tend to check meta before MO and as there wasn't a comment here saying that the question had been posted then I assumed that it hadn't yet appeared on MO and so left my comment. I do think that if a question that has been discussed in advance on meta is posted then a link should be given here. As no one else did so, here it is.)

26.
In one of the earlier posts in this discussion, I noted Littlewood's demonstration that a mathematics dissertation could be two lines long. In the August-September 2014 Monthly, page 618, Don Zagier improves on Littlewood's effort. First, he notes that the statement of the theorem could be in the title, reducing the length of the thesis itself to one sentence. Second, he replaces Picard's small theorem, Littlewood's choice, with one that he says is better-known and easier to prove. The title is "Bounded entire functions are constant", and the proof is "If $f(z)$ is bounded and entire, then Cauchy's Theorem implies that $$f'(a)=\lim_{R\to\infty}\left({1\over2\pi i}\int_{|z|=R}{f(a+z)\over z^2}\,dz\right)=0$$ for every $a$ in the complex plane, so $f(z)$ is constant."
27.
Gerry, the displayed formula turns this into three lines. I'd rewrite it so the part after the displayed formula appears before it, and that would reduce the thesis into two lines. But we already had that result.
28.
Use a smaller font, then, and put the math in the text, not separately.
29.
Joel,

using a small font doesn't count. The entire bible was inscribed onto pinhead.
Surely a 100 pages long thesis could be inscribed on a fraction of that pinhead using similar font settings.