A Question about Doctoral Theses in Mathematics [closed]

Question

This is most definitely a soft question, which I'm sure may get some negative attention, and perhaps even be voted closed. However, I genuinely would like to generate answers on this matter as it concerns anyone that decides to get a doctorate in mathematics. (Since I believe this is the best forum to do so, here goes...)

Question: Does a doctoral thesis in mathematics have to contain more than an abstract, a proposition and proof of a new and exciting result?

I strongly believe that brevity is beauty to mathematicians and see no problem with a thesis that contains a lengthy abstract (~350 words) to generate enthusiasm and then jumps right into the heart of a novel and far-reaching result. The proof provided is extremely condensed in the spirit of Zagier, leaving almost all details that can be found elsewhere to be found elsewhere, in the works cited. If the result has merit, I see no reason why it can't be submitted or published in such a manner. To support my stance, I offer the following condensed theses:

1. David Rector, "An Unstable Adams Spectral Sequence", MIT (1966), 9 pgs.
2. Burt Totaro, "Milnor K-Theory is the simplest part of algebraic K-theory", Berkeley (1989), 12 pgs.
3. Herman Buvik, A New Proof of Torelli's Theorem, NYU (1962), 12 pgs.
4. Eva Kallin "A non-local function algebra" Berkeley (1963), 13 pgs.
5. Edmund Landau "Neuer Beweis der Gleichung $\sum_{k = 1}^{\infty} \frac{\mu(k)}{k} = 0$", Berlin University (1899), 14 pgs.
6. Barry Mazur, "On Embeddings of Spheres", Princeton (1959), 26 pgs.
7. John F. Nash "Non-Cooperative Games", Princeton (1950), 27 pgs.
8. Kevin Walker, "An Extension of Casson's Invariant to Rational Homology Spheres", Berkeley (1989). 29 pgs.

Although I'd appreciate to hear from anyone worth giving his two cents, I'm specifically eager to hear from those senior members of the community.

Thanks!

Answer 1

In the end, the doctoral thesis has to contain enough to be passed by the readers (and to satisfy any auxiliary rules of the institution to which it is submitted). While a very slim thesis is unusual, if the results are good and the writing is comprehensible, I don't see that slimness is necessarily bad.

On the other hand, there is a tradition of including a certain amount of expository materials in a thesis, and I think this is useful, both for the readers (who may not be as familiar with the area as the student and advisor, but who have to form a judgement on the thesis) and for the student writing it; it is good to practice writing clear expositional and motivational material, since this is similar to the kind of material that is required in introductions to papers, in grant proposals, in colloquium talks, and all the other things one has to write in one's career that will be read or listened to by people who are not as expert in one's area as oneself.

It is also quite likely that you will be sending a copy of the thesis, or at least a provisional version, to your letter writers, and so any expository or motivational material that will enhance their appreciation of the thesis is certainly worth including.

Overall, bear in mind that while a brevity can be a good thing, if it degenerates into opacity, that is bad.

Answer 2

Eva Kallin is a colleague of mine from Brown, and she always expressed pride in the brevity of her thesis. In all my professional years, I never heard anyone disparage that dissertation. It seems to me that generally, mathematicians envy those people who have produced unusually short theses.

Remember, no mathematician is expected to show encyclopedic knowledge of mathematics, especially in the thesis that’s submitted.

Answer 3

[Disclaimer: I’m a fairly junior member of the community — in my 3rd year of postdoc-ing.]

Yes, there are certainly plenty of excellent short theses in maths. There’s nothing necessarily wrong with a short thesis; it doesn’t have to contain any more than what you say.

However, I’d suggest being quite careful about the issue. In a small thesis based tersely around a key result, that result has to bear a lot more weight than if it were part of a larger thesis along with a bit more general exploration of the area. If I go to a restaurant and the portions are tiny, then that can be fine, but the food had better be really good. And this isn’t just silly prejudice: a longer thesis really can exhibit aspects of a candidate’s knowledge and capabilities that a single tersely-proved result, even an excellent one, doesn’t attest to.

Finally, the audience of your thesis is your PhD committee. If the certain faculty member is on your committee, or close to people who are, then it’s not good to antagonise them. What will be read by a wider audience is determined by what’s on your webpage, on the arXiv, or in journals. If you make sure to make your perfect, terse exposition the more readily accessible one in these ways, then perhaps it may be easier to bite your tongue and write the thesis that will be more acceptable to your thesis committee.

Answer 4

In Littlewood's "Mathematician's Miscellany" (fairly recently reissued as "Littlewood's Miscellany"), this exact situation is discussed. He says that, yes, even a two-sentence dissertation could be acceptable. He says that Picard's great theorem (iirc) with a one-sentence proof is an example.

I $love$ this book and highly recommend it to anyone with any sort of interest in mathematics.

Answer 5

I'm not 100% sure that you yourself are the author of the thesis rather than the supervisor or some other interested party, but since you are offering to send it out by email I am probably safe in assuming the former.

In either case, I think you should consider carefully whether this is the best place for you to air a dispute with another faculty member. It seems likely that involved parties could recognize themselves or you and that this could be seen as unprofessional behavior.

In any case, what do you hope to achieve by asking this question? You are clearly hoping that M.SE users judge your own case, of which they know only your side, on its merits, rather than simply start a theoretical discussion about thesis length. Even if many or all of the answers agree with you, will this really get you any further in your dispute? The responses of strangers on the Internet are not universally considered to be valuable. If they are helpful for mathematics or programming questions, that is partly because they involve questions of information and verifiable fact.

You should also be aware that comparing your work to that done by a list of great mathematicians might not help your case either. This is frequently done by cranks. As you know, those theses might have been short because they were great, but they were not great because they were short.

Maybe a better approach would be to find some faculty member whom you know reasonably well and trust and try to get them to see your side of the story, and advise you on how to go forward. If you can't find anyone to take your side, you need to ask yourself is there is a reason for that.

If your result is great it will be recognized as such, regardless of how it is written up. It would be a shame if the matter became a distraction from your hard work, or if it led to the perception that you were arrogant or disrespectful to your senior colleagues.

Answer 6

I'm not an academic, but I got a PhD in (applied) maths over 25 years ago and I do regard myself as a mathematician (among other things).

I've always thought that the main point of a PhD is to prove that the author can do original research. That was certainly why I was always interested in hiring people with PhDs when I used to work in the banking industry, because I wanted to hire creative people who could have their own new ideas. So my view is that in principle, a short PhD which contains original research would be absolutely fine :-).

However, I think the problem with a short PhD is a practical one. The university that awards the degree needs to be certain that the author is responsible for the original research in their PhD. The less there is in the thesis, the less there is to discuss and test in the viva, so establishing that the person is the true source of the exciting new result is much harder.