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This question stems just from curiosity, as my work is not concerned with mathematics.

What happens if a student doing a PhD in mathematics submits a thesis with a proof of some theorem and the reviewer finds fatal flaws in the proof?

I don't mean small typos but flaws of greater caliber, whose fixing leads to the conclusion that the conjecture one was convinced of proving is completely wrong, or, if gaping contradictions in the proof are found. Can one obtain a PhD with such a flawed thesis?

I assume that in other areas of research even if some results are proven wrong, the PhD can still be awarded on the basis of other results that are present in one's work.

Note, this question differs from this one as here we assume that the thesis has already been submitted and is under review.

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    @Justus That seems a bit extreme. Kempke's flawed proof of the four color theorem had definite value, and even contained some of the ideas that led to the eventual proof a century later. A "good" flawed proof is often a perfectly valid proof of a weaker version of or a special case of the theorem that it attempts to show. I agree that a fundamentally flawed proof can't be awarded a Ph.D., but it is quite plausible that the bulk of the work could still be salvaged. – John Coleman Jul 31 '17 at 2:31
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    You may want to read William Gasarch's blog post A Mathematical Urban Legend (2010). – Rodrigo de Azevedo Jul 31 '17 at 12:44
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I'm aware of a thesis that unravelled at the oral defense: one examiner found a small error and it turned into a big error that invalided a substantial part of the thesis. The candidate was required to make major revisions. (The party scheduled after the defense was promptly cancelled.)

I'm also aware of a thesis that was downright rejected by the external examiner before it reached the oral defense stage. The candidate also has to make major revisions.

These occurrences are very rare: most programs have various milestones to satisfy before a thesis can be submitted, and these roadbumps are there precisely to avoid this kind of mess (which makes everybody look bad, especially the supervisor and the supervisory committee.) It's rare but it does happen, and on both occasions the candidate dug in, cleaned up the work, resubmitted some months after and eventually graduated.

(One thesis was in physics, the other in math.)

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I know of two cases where a big flaw was found in a dissertation when the student was in the home stretch. In both cases the student needed to fix it completely and resubmit before defending. Fortunately this didn't create problems with job offers in these two cases. I have no idea what the programs would have done if a student had already accepted a job contingent on receiving the PhD.

I also know of a few situations where a dissertation was scooped very late in the process, which is similar: if the work isn't "new mathematics," then it's back to the drawing board.

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    What kind of (mathematics) committee insists it must be new to that extreme? The Ph.D. is a certification that the candidate can perform modern research in an effective and independent fashion. That's just as demonstrable with independently conducted research that coincidentally happens to produce results someone else narrowly beat them to. And if the methods are distinct then that itself carries intrinsic value. It's not a desirable situation for establishing yourself on the job market, but it hardly seems like a "no Ph.D. for you!" thing. – zibadawa timmy Aug 1 '17 at 0:08
  • That's the way it works for a PhD because that's the way it works for publication. Sometimes a novel method is enough, but often it isn't. – Elizabeth Henning Aug 1 '17 at 0:28
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    Certainly that wouldn't be the decision of every committee. Usually such a situation can be made to work without the student being left truly high and dry. – Ben Webster Aug 1 '17 at 2:09

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