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I have got an article the making, and presented it to my PhD advisor one month ago. The article emerged from my master thesis at a different university.

A week ago, he told me the draft is good, albeit difficult to understand, but he remarked that it contained too many new ideas to be published at once. The problem is that no journal will be able to find a reviewer. Furthermore, I am just some unknown PhD student to them, so there won't be a "rubber-stamp" for me.

The draft contributes to an active mathematical research area which leads previously unrelated topics together - in particular, very different mathematical communities. My draft contributes by another bridge between communities. I am convinced that people will find the result noteworthy and interesting, but I also agree that the combination demands (basic) knowledge from many different mathematical areas.

It is a disturbing but possibly realistic perspective that researchers keep the same old soup at low temperature on the cooker. But isn't that too pessimistic? There are so many mini results published, why should I cut down on at least trying?

Are the concerns of my advisor well-founded in practices of the mathematical community?

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"too many ideas to publish at once" - does he mean you should split it into a series of (more focused) papers? –  Ran G. Jul 15 at 22:52
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I have to disagree with the premises of this question... conceivably the advisor was using some euphemistic way to impart some message? Basic knowledge is not such a rarity as people like to pretend. @PeteLClark's answer's content is constructive. Don't let yourself believe that serious mathematicians are low-energy ignoramuses outside their own tiny bailiwick. Good journals can find reviewers, don't worry. –  paul garrett Jul 16 at 0:08
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If it's an "active mathematical research area", that implies a significant number of researchers working in it. That's your pool of potential reviewers. There doesn't need to be hundreds of them, just so long as there are a few and an editor can find them. –  Nate Eldredge Jul 16 at 0:49
    
Does your adviser understand your paper?! If so, then a journal should be able to find referees (unless your adviser is a #1 genius, in which case you should probably do as he says). If he does not understand your paper, then you should ask him to suggest someone who will, and then ask that person for publication advice. –  A.G. Jul 19 at 4:26

3 Answers 3

No, in my opinion as a (midcareer verging on senior) mathematician, there is no such thing as "too many new ideas to be published at once". This is a curious reaction.

Whenever a student asks a "My advisor said....[something strange]" question here (or elsewhere), I have to wonder how deeply to engage in the possibility that the student somehow misunderstood the advisor. Unfortunately misunderstandings between students and advisors are amazingly common...even when both parties are "good people" in every sense of the word. More than a decade after graduating, I still remember spending several weeks of hard work on certain things my advisor asked me to do. When at last I would go back explaining how hopeless it seemed, it most often turned out that there was some miscommunication: I didn't work on the problem that he had intended, or I hadn't read the right paper. [I remember wading through Katz's Rigid Local Systems because of a suggestion made by my postdoctoral supervisor. Being a postdoc I was more savvy than a PhD student, so I probably spent at most a week before I went back to say "Really? This is relevant to what I'm doing??" only to learn that, no, that was not the book of Katz he was talking about. Still, I'd like to have that week back...]

I would encourage you to consider the comment "the draft is good, albeit difficult to understand". I'm guessing that is meant to be independent information from the remark that your paper contains "too many new ideas". Some key questions:

  • Was the draft prohibitively difficult for your advisor to understand? People [students, advisors, referees...] say that writing is hard to understand in two very different circumstances: they may literally be criticizing the writing style. But it is also quite likely that they are trying to say -- in a way which saves face for them and for you, but is in fact rather unhelpful because of that -- that they gave up before they could tell whether your arguments were correct, and they would only be willing to put more time into a better-written draft. You deserve to know whether your advisor vouches for your work.

Related:

  • Does your advisor have the expertise in the disparate areas you are pulling together? If not, he is really not the right person to be asking about this. People sometimes seem to think (or more likely, to hope) that if they have a paper on "number theory" then I will understand it, and that if the paper references some work or lecture notes of mine then I will really understand it. No way. I often receive papers to referee which are about several things at once, one of which is part of my expertise but one or more isn't. I like to learn new things, so I'll stretch to a certain point, but beyond that I just decline to referee the paper on the grounds that I'm not qualified. If I can, I direct the editor to someone who is qualified; if I can't, I apologize for not doing that, but in no case do I intimate that no one is qualified to referee the paper. How could I know that?!?

The idea that your paper is simply too ambitious is really a poor one. At most it means it is too ambitious for your local mathematical community, and if you hear that as a reaction to your work which is otherwise said to be "good", it's a sign that you need to find a larger pond.

I am convinced that people will find the result noteworthy and interesting, but I also agree that the combination demands (basic) knowledge from many different mathematical areas.

Then you should find people who feel that the result is noteworthy and interesting, ideally those who understand all of it, but even those who understand some of it and can be supportive could be helpful. If you think your paper is correct and reasonably (even if not perfectly) well-written, why not submit it to the arxiv? If your paper draws together several different things, then try showing it -- via emails, for instance -- to people wom you know to be experts in at least one of those things. They can tell you whether they are qualified to understand the entire paper, and if not they can (perhaps) tell you who is.

It is a disturbing but possibly realistic perspective that researchers keep the same old soup at low temperature on the cooker. But isn't that too pessimistic? There are so many mini results published, why should I cut down on at least trying?

Yes, that is too pessimistic. If you've done something valuable and technically difficult, it will be publishable. The level of technical difficulty may make the reviewing process more lengthy (it should; you are aware that you don't want rubber stamp reviews of your math papers, right?). On the other hand, in mathematics "technical difficulty" can be a selling point: if you have done something that is broadly valuable but that few people (or no one but you) would have had the acumen to pull off, then you have done something very impressive and valuable indeed.

Good luck.

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Among your other good points, the construct of the template of "advisor said <something weird/unintelligible/...>" is very much worthwhile. Maybe this deserves a whole SE website to itself... or at least a tag? So then part of the community's edits would have to be the addition of that tag, since the questioner would presumably not know that that was a huge part of the issue, etc. Like the fact that the special disadvantage of the inexperienced is to fail to understand the potential advantages of experience. And there's only so much fun to be sucked out of self-referential stuff... –  paul garrett Jul 16 at 0:50

As a mathematician who works squarely in between fields, there are some challenges which can make getting such papers accepted more difficult. However, all that means is that you have to work a little harder at writing clearly and accessibly. Think hard about who is going to read the paper and give them what they need to understand the paper.

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As an editor, I once got a submission that combined two very distant areas of mathematics. That was no problem; I just sent it to two referees, one in each area, asked each one to referee the part in his area, and assured each one that I had another referee checking the rest of the paper. Of course, once both referees reported that their part of the paper is good, I still had to make sure that the parts fit together properly, but that was easy enough for me to do by myself.

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