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I've recently come across a math PhD thesis of a student from a prestigious US university in which he just computationally checked, up to a certain bound, a conjecture, without coming up with any new ideas. To me, this would be insufficient to get a PhD, as there are no new insights. I was talking about this with a collegue, who showed me a phd thesis in which the main thesis was a not well-supported conjecture. Hence the question:

how common is for successful PhD theses in math to have "little to no results" in them? Do you know any further examples (of course, without mentioning names, unis, etc)?

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    Question: what subject was the thesis in? It could be computational mathematics, and the new, novel idea could be the algorithm used to check the cases. – Johanna Mar 1 '15 at 15:01
  • @Johanna The subject wasn't computational math. – user31119 Mar 1 '15 at 21:44
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    I've been wondering this too. As a Ph.D. student about to finish, I feel like my "results" are quite insignificant. My advisor seems to think what I already have is enough for a these, but I think it's pathetic and similar to the thesis you've described. – Felix Y. Mar 2 '15 at 3:26
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    @user31119 or maybe it was number theory? I know of a number theory thesis where the student improved upon an algorithm for finding primes, which sounds like it could be a computer science thesis as well. I'm sure there are quite a few cross-disciplinary theses, where the student starts out in one field and then achieves something in the intersection with another field. It's not necessarily a boring thesis just because it left the original field. – Sumyrda Mar 2 '15 at 6:52
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Not all theses are groundbreaking. This is true in math as in any other discipline. Oftentimes, an adviser has a hunch that going in a certain direction might yield something interesting, but that turns out to be wrong -- sometimes because the student didn't work very hard, or simply because there is nothing there. For example, in math, the original conjecture might simply have been wrong; it may have been correct but too hard for even a good student to prove; or, maybe most frequently, it is true in some cases for which it was proven but these cases end up looking rather insignificant.

So, from the perspective of a department, what do you do? The student did work, his work ethic was average, nothing of great significance came of it at the end of 5 years, but it wasn't for lack of trying. Some incremental progress was made, but nothing that looks particularly impressive. Do you kick the student out of the program after 5 years? Do you let them work on this for another 1 or 2 years with uncertain prospect? Or do you simply declare victory, bury the topic, and make sure the student graduates in a reasonable time and everyone moves on with their lives?

You will find that this last option happens surprisingly often. The number of real breakthroughs is, after all, rather small.

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