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This is a bit of a long story. I am a graduate student working on mathematics and the topic I have been working on is not directly within my advisor's expertise (pretty closely related to though). My thesis basically consists of two parts which deal with two related but independent problems. At the beginning when I chose these two problems to work on, I was kind of discouraged by my advisor as he thought these problems despite interesting did not seem reachable. But actually, when I decided to work on these problems, I already had some rough ideas on tackling the problems.

Anyway, the first part was basically done in last summer/fall which I also presented my proof orally to him. For the second part, he was once kind of skeptical of my proofs, however, his confusions and doubts seemed to be resolved through our conversations, email exchanges and my improvement of the exposition. I sent him my first draft in April (I was supposed to submit my thesis in June). In late June, when I had to submit my thesis, he said he had not really finished reading my thesis.

After my thesis was submitted, he told me that my thesis was very badly written which made him not even able to follow my work. But wait, he had two months to take a look at my thesis but he had not said a word about that every time we met. Anyway, my thesis defense was thus delayed for two months and in these two months I was constantly asked to make changes big or small on the thesis. At certain point, my advisor seemed to feel my thesis was in good shape and I should arrange for the defense.

Alright, I did the defense in August and it went well. But recently I received my thesis feedback report in which my advisor suggested that I should remove the whole second part from my thesis because he thinks that my proofs/arguments presented there are not solid enough. I always appreciated when he could point out specific mistakes/confusions I made in my thesis. But it seems that after all of these have been resolved, he is still not convinced that there is no gap in my proofs. I really do not know what to do now.

Should I argue with my advisor or should I simply follow his request to remove that chapter and get my degree safely?

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    I suggest using paragraphs. Anyway: what is your next career goal, and what part of the world are you in? – cag51 Sep 21 '20 at 19:32
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    Is he willing to approve it and let you graduate without the chapter? Is the rest sufficient? – Buffy Sep 21 '20 at 19:44
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    Have you asked a near identical question here before? – Bryan Krause Sep 21 '20 at 21:45
  • @cag51 I still hope and plan to continue my research in the same field. – user129757 Sep 22 '20 at 16:47
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    What did the other committee members say/write after the defense? You might end up in a situation where they do not want to approve the thesis without the part that your advisor wants removed. (Which is why in a sane system, the committee discusses changes and makes a single feedback report of what to change.) – UJM Sep 24 '20 at 17:48
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It is very likely in your interest to follow your advisor's advice. The timeline you describe suggests your advisor has been highly irresponsible (or impacted by the current pandemic) about providing timely feedback, but this is not a reason not to do what they recommend.

In academia, you will be judged on your publications, not the content of your thesis. Removing content from your thesis to please your supervisor will not necessarily prevent you from publishing that work, so the downsides of a shortened thesis are limited.

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    We were in the middle of a pandemic, so "highly irresponsible" really does not seem accurate. – Andrés E. Caicedo Sep 22 '20 at 4:23
  • I do want to continue in academia. It seems to me the content of my current thesis would affect my applications. Also, if I do not even try to argue with my advisor but simply follow his request to remove that part of my work away from the thesis, does it mean (at least to him) that I admit my proofs presented are problematic? – user129757 Sep 22 '20 at 16:30
  • @user129757: When you apply, you can link the extended/updated version of your thesis rather than the version you submitted at your defense. PhD students link their unpublished preprints all the time; this is no different. – darij grinberg Sep 23 '20 at 15:14
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Congratulations on your (presumably) successful defense, and sorry to hear about the timing issues.

Two important points are not mentioned here that may affect what you should probably do:

  1. Is the advisor the only remaining gatekeeper to you getting the degree?

  2. Do you wish to publish or pursue this topic otherwise, and are you interested in improving your mathematical writing?

Frankly, given all the delays (regardless of what caused them), if you just need your (seemingly) overworked advisor to say "yes", and they are prepared to do that if you remove a chapter, my first reaction is just to get that done and then figure out what to do with the rest!

If you need to still satisfy other committee members (may or may not be relevant in your geography/institution) and they don't agree, then welcome to the world of academia, where we end up in that boat all the time with reviewer feedback for submitted articles!

If you do want to work in this area and continue in academia, then don't stop there however. Regardless of what prompted the initial delays, advisors don't throw up their hands in frustration at whole chapters, after a successful oral defense and some back'n'forth, out of sheer laziness. You probably have something interesting there, that came across strongly in the oral defense, but continues to not be written clearly enough to hang together for a reader trying to make sense of it. You have doubtless had the experience of reading published papers where you sort of follow each paragraph but you're banging your head trying to make sense of it all. It sounds like your advisor might be in the same boat with your chapter, and has now moved to pragmatically trying to push you through.

If that is the case, and you do want to continue with an academic career in mathematics, you should probably work with someone to continue to polish this and therefore learn how to write more clearly. Perhaps not your advisor (since it seems hard for you and them to achieve this, based on experience), but perhaps someone else? And if you do so, there will be absolutely nothing wrong about having more results than in your official thesis, and/or a richer preprint or 2 preprints instead of one to show for it.

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    "advisors don't throw up their hands in frustration at whole chapters, after a successful oral defense and some back'n'forth, out of sheer laziness." False. Most don't. Good ones don't. But it does happen. – Anonymous Physicist Sep 22 '20 at 0:52
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    @AnonymousPhysicist I'd give them the benefit of the doubt: Maybe they have realized that (i) they are not satisfied with the chapter, (ii) they currently don't have sufficient time to help the student bring it into shape, (iii) the student nevertheless is ready to graduate. In such a case, they might actually be doing the student a favor by asking them to remove the chapter and letting them graduate with the other chapter. – Roland Sep 22 '20 at 6:03
  • 1. There are other committee members but my advisor is certainly the gatekeeper. 2. I wanna continue my research in this direction as well as to improve my current writing. I also agree that working myself or with someone else to improve the quality of my writing is a good idea. But what also concerns me is that if I do not even try to defend for my work, my recommendations from my advisor or other committee members in future applications (if I ask for them) will be weakened. (actually, I doubt if my advisor will write me a good one based on what we've been thru last few months....) – user129757 Sep 22 '20 at 16:44
  • Good answer, but the problem with contradictory reviewer feedback is somewhat overhyped: Usually, in those cases, editors expect you to be the judge in whose advice you follow. Theses might be a bit different, but in general the rest of the committee (i.e., the non-advisors) don't care much about the thesis to begin with and thus won't hold it up. – darij grinberg Sep 23 '20 at 15:16
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Keep in mind that you can always post an updated version of your thesis on the arXiv (or, if this wasn't mathematics, at various other places on the internet). Thus, you shouldn't pay much attention to what thesis you submit at your defense; you can always "beat" it with an arXiv revision. If your advisor has doubts about a chapter while believing the rest of the thesis to be already sufficient, there is nothing easier than just submitting the rest to your university and then posting the full version on the arXiv. (Just make sure to document that the full version is an update.)

Independently, if you haven't already done so, I'd suggest you get some outside opinions (apart from yours and your advisor's) on whether that fateful chapter is actually rigorous or not. This is probably useful anyway, seeing that your advisor is a stranger to the topic; it would get you familiar with people working in the topic.

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