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I'm currently working on my PhD thesis. I have several suggested problems to work on, some of them are very similar to some problems that my advisor have worked before and published already, either in his thesis or papers. Basically, the main difference is in the dimension of some singular sets (his works are mainly on the isolated case, but I'm working on a case with a far more hairier, non-isolated singular set), which we were unsure if the argument would hold but it seems that the adaptations I've made were fine.

Not that the nature of the problem matters, but the approach I'm making worries me. It seem to me that if there would be a 'railroad' to prove the results I'm working on, it would be the same path that he followed to write his own results, with different objects. That's the way I've been advised to work, and it's been producing results.

Is this a reproachable approach? Of course, there is the problem of using similar introductions (and in that subject I've read this previous question Does this qualify as "self plagiarism" or something? , only one that got close to my problem) sinde the objects being studied by me and that has been studied by him were so similar.

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    $\begingroup$ There is some value in checking that a given argument extends to a slightly more general setting. You can not expect it to be published in a particularly good journal, but your first paragraph makes it sound like the research is legitimate. You can always show it around to experts in the field to see if they view it "too obvious to publish". $\endgroup$ Apr 25 '16 at 20:35
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    $\begingroup$ "plagiarism" is the term used if you don't cite your sources; if you do, don't worry. $\endgroup$ Apr 25 '16 at 20:47
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    $\begingroup$ Of course it would be better if you could come up with a brilliant new method that blows all these problems away. In the absence of that: most research makes incremental progress by adapting existing tools to a slightly different situation. $\endgroup$ Apr 25 '16 at 21:05
  • $\begingroup$ Highly relevant: youtube.com/watch?v=IL4vWJbwmqM $\endgroup$ May 1 '16 at 13:01
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What you describe seems to me to be a normal mode of mathematical progress, and I would urge you simply to carry on! Ride that train as far as you can.

It often happens that someone's mathematical results can be improved or generalized in various ways, and when this is possible, it is mathematically desirable that the generalization be undertaken well.

You may be worried that the value of this work is less than some other totally original work. If the generalizations are routine, then indeed that may be true. But from what you say, this doesn't seem to be your case. Many generalizations are not routine and such work is definitely worth doing.

Finally, let me caution you to guard yourself against a certain mistake that sometimes undermines motivation for a young researcher. Namely, it often happens in mathematical research that we begin in a state of terrible confusion about a topic; as research progresses, things only gradually become clarified. After hard work, we finally begin to understand what is the actual question we should be asking; and then, after fitful starts and retreats, we gain some hard-won insight; until finally, after laborious investigation, we have the answer.

But alas — it is at this point that the crippling illness strikes. Namely, because the researcher now understands the problem and its solution so well, he or she begins to lose sight of the value of the very solution that was made. The mathematical advance begins to seem trivial or obvious, perhaps without value. Having solved the problem so well, the mathematician becomes a victim of his or her own success. Because all is now so clear, it is harder to appreciate the value of the achievement that was made.

Please guard against this disease! Do not denigrate your achievement simply because it seems easy after you have made it. In many mathematical realms, the actual achievement in research is that certain issues and ideas become easy to understand. Please look upon the ease of the answer at the end as part of the achievement itself, and think back to the initial state of confusion at the beginning of the work to realize the value of what you have done.

So please carry on and ride that railroad as far as it will take you.

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    $\begingroup$ I remember being envious of carpenters and artists, who after an achievement can point at what they've done, its value obvious; while in mathematics our very achievements can undermine our perception of their value. $\endgroup$ Apr 25 '16 at 21:44
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    $\begingroup$ I heard once that Gödel worried briefly after having proved his incompleteness theorem that his accomplishment might be little more than another simple paradoxical play on words. Of course that is not how we look upon it today. $\endgroup$ Apr 25 '16 at 21:49
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    $\begingroup$ Let me add, as a corollary to the point I make in the post, is that we shouldn't be impressed with a difficult mathematical argument simply because it is difficult. Rather, let us criticise difficult arguments for being obscure and try to improve them. $\endgroup$ Apr 25 '16 at 22:42
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    $\begingroup$ I remember being envious of carpenters and artists, ... our very achievements can undermine our perception of their value. - I don't see how this phenomenon of devaluing one's work after completion is foreign to artists (and possibly carpenters). $\endgroup$
    – Kimball
    Apr 25 '16 at 23:24
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    $\begingroup$ " In many mathematical realms, the actual achievement in research is that certain issues and ideas become easy to understand." --- I really like this insight. $\endgroup$
    – Nik Weaver
    Apr 27 '16 at 0:43
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One major aim of a PhD is to study a problem so intently that suddenly, after much hard work and perseverance, the solution becomes obvious - or at least that it becomes obvious what the approaches to the problem would be and why one of these approaches is likely to be superior to the other approaches. Be aware that everyone has good ideas - and groundbreaking approaches tend to build on earlier approaches. My PhD was "simply" the combination of three existing approaches in a novel way. But to get to the point where it was obvious that these three things together were what was necessary or would produce a solution took 2 1/2 years' work.

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Just do what your adviser tells you to do and don't worry. Adviser has to approve your thesis before you defend or publish it. With his/her approval who can blame you in plagiarism from your adviser?

EDIT. Concerning all other sources (other than your adviser) you should do all you can do to make sure where the ideas come from and give proper references.

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    $\begingroup$ Be sure to cite the prior work prominently in any oral or written presentations of your work. Essential! $\endgroup$ Apr 26 '16 at 1:18
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    $\begingroup$ It can also be valuable for you to outline what parts are your argument are similar to arguments in the literature and what arguments are new. This not only served to give the originators of ideas proper credit, but also can help a reader understand the nooks and crannies of the new argument. Finally I believe this answer is meant to point out that your advisor might be best suited to help you answer these questions. That is certainly true, but remember it will be your name on the work. You have to be satisfied with how work is cited, in the same way you have to stand behind the math. $\endgroup$ Apr 26 '16 at 6:15

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