Though I don't have a graduate degree, I have original (and rigorous!) mathematics and have been cold-emailing some professors. Unfortunately, none of them want to collaborate online with me.

I really would like to get my research accepted. What can I do?

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    The standard way to get your research accepted is to submit it to a peer reviewed journal and have it accepted there. That can be done without the help of a professor (although admittedly it’s not easy). And if you need significant assistance from a professor, the standard way to get that is to enroll in a graduate program. So those are the main two ways to “break into mathematics”.
    – Dan Romik
    Commented Nov 6, 2021 at 21:45
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    original rigorous maths without a graduate degree in 2021?
    – BCLC
    Commented Nov 7, 2021 at 10:32
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    @BCLC Math research prior to a master's or PhD is not that uncommon. However, it is true that successful undergrad/etc research usually involves a mentor.
    – Kimball
    Commented Nov 7, 2021 at 15:35
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    Contacting present professors is very unwise. Contacting future professors (a small percentage of present grad students) would be a bit less unwise. You may be underestimating the demands on people's time in academia. An assistant professor I once collaborated with was ignored by tenured professors at one of the "big name" research universities because he had not yet built a reputation on his own (by publishing highly cited papers without his doctoral advisor's supervision). He had a PhD from MIT and his emails were not returned. Please consider being realistic. Commented Nov 7, 2021 at 21:47
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    As @RodrigodeAzevedo comments, people with stellar credentials already have difficulty with cold-calls and publication and so on. That is, to be realistic, you are competing with better-credentialed people who already are having substantial difficulties getting what you're wanting... Not that you shouldn't try, but do be aware of the steep uphill here... Commented Nov 7, 2021 at 23:26

7 Answers 7


Read enough math papers so that you have a sense about how they are written up. Then write your own, emulating the style. Then submit it to a journal. You don't need any degree for this.

You will get feedback. If the work is not very good, it will be immediate feedback and you need to think about what you are doing. If it is good, then, after a while you will get a reviewer report suggesting changes.

You don't need collaborators, but finding someone to read your work and give you some feedback is helpful. If you have an undergrad degree, or are an undergrad, one of your professors can probably help.

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    I think this answer is good, but it is really hard to believe that someone can learn enough about proper math writing without mentoring of some sort to be able to publish. Commented Nov 7, 2021 at 20:25
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    @DenisNardin I had a colleague who started publishing quality papers in his early twenties without any help — it took me a couple of years worth of mentorship to get to his level. Some are naturals, which I have witnessed a couple of times already. Commented Nov 8, 2021 at 5:56
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    @Buffy That's actually a terrible example: Ramanujan was unable to get other mathematicians to take him seriously until Hardy and Littlewood gave him mentoring in how to write. Still, I do not disagree with you, it's just that it's a lot harder than you're making it look. Commented Nov 8, 2021 at 15:01
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    @Buffy That's one absolutely extraordinary example. Surely the topic of statistics is not foreign to the mathematical reader. One person's success, out of billions, over centuries, does not change the fact that the probability of OP having made a groundbreaking discovery is effectively almost zero. Every professor evaluating the prospects of engaging OP is acutely aware of this, which is why nobody wants to talk to them. Correcting bad ideas and guiding students is the job a professor is paid for. OP wants that service for free - completely on the merits of their "amazing" research. Unlikely.
    – J...
    Commented Nov 8, 2021 at 19:49
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    @Buffy No, I'm just saying that we shouldn't expect events to be low probability events simply because they occasionally happen. Nobody will be faulted for giving the benefit of the doubt to the more probable explanation - in this case, that OP is almost certainly not a mathematical genius and is, quite likely, sitting on a paper that is worth nothing.
    – J...
    Commented Nov 8, 2021 at 20:38

Possibly you don't want to really ask for "collaboration" as much as "advice" or "feedback". Advice and feedback are "limited" responsibilities, while collaboration is an on-going thing. People might be willing to give advice but not make any long-term commitment.

Also, keep in mind that most professional mathematicians, by their own design, are already "fully-booked" for the foreseeable future, apart from possible large surprises which would justify changing their schedule abruptly.

That is, they already have many on-going projects, as many collaborators as they'd want (if not too many!), as well as other obligations. So, imagine that you are asking a person who is already extremely busy to do something extra. Not that that's impossible, but "it's a big ask". :)

EDIT: ... and, anyway, as in other comments and answers, you don't have to have collaboration of a professor to attempt to publish (in the old sense of "in a refereed journal") your work. Yes, the stylistic expectations, and judgement about level of innovation and so on, will be easier for an experienced person, but in that regard all you really need or want is "advice" rather than "collaboration".

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    Is there a latex template I can use for simple papers?
    – Wakem
    Commented Nov 7, 2021 at 16:45
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    On arXiv, you can see the (La)TeX source as well as the typeset version... Choose a smallish paper and see what's involved. Yes, starting from scratch it'll seem intimidating, but it's just a one-time thing. There's also no necessity of understanding the inner workings of TeX, etc., any more than we reeeeally have to understand how networked computers work to use them. :) Commented Nov 7, 2021 at 17:38
  • @Wakem Depends on the journal you plan on submitting to. I know that there's a template for IEEE journals, for instance (though I doubt that the IEEE would be interested in publishing a paper on pure mathematics unless it was directly related to their areas of interest, like some computer science stuff might be). You might be able to ask the editors of a specific journal if they've got a LaTeX template that they'd prefer for their submissions to use.
    – nick012000
    Commented Nov 8, 2021 at 7:06
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    @nick012000 In my experience there is no expectation that math authors use a journal's preferred template before the final stages of publication. Commented Nov 9, 2021 at 2:01

Against some of the other answers, I will give a more pessimistic view here.

It will be almost impossible for you to get published from outside academia. I'll outline a few reasons.

  • Nothing prevents you from submitting your work to a journal; an academic affilitation is not a requirement. That said, to get published you need to impress one or more experts (the editor(s) and the referee(s)). Unless there is some crazy exceptional idea in your work, badly written mathematics will fail to impress. Which brings me to the next point.

  • It usually takes more time than a full Ph.D. for a mathematician to start writing mathematics decently. That is, years of practice and mentorship; a big part of supervising graduate students has to do with getting them to write properly. So I would deem it impossible that whatever results you have are written in a way that a referee will find acceptable. Refereeing a paper is a lot of work, and referees get understandably upset when papers are written poorly.

  • Without mentorship and contacts in academia, it would be really hard for anyone (not just you) to assess the quality of your own research. Most non-trivial research benefits greatly from discussions with other experts. And it is a common experience for all (most?) mathematicians to have a "brilliant" idea, that immediatly becomes stupid when we explain it to another mathematician. We have all thrown some "earth shattering" paper to the trash when we finally realized it was worthless.

  • Which brings us to the issue of contacts. Even for an absolutely brilliant mathematician, it would be hard to be heard from outside the community. Most famous case being that of Ramanujan. His first attempts to get in touch with British mathematicians ended up in rejection precisely because his (brilliant) arguments were poorly written. And some were wrong, as most of his results on prime numbers were. Hardy's wisdom and mentorship were required to make Ramanujan a mathematician. Another famous example is Fourier, whose papers were (rightfully) rejected even though they had brilliant ideas.

The only chance you have to get an inside is to hopefully get in touch with someone who is a direct expert in the (sub) area your results are in. That said, we are all used to receiving papers from cranks; the only way I would (maybe!) take a closer look to a paper sent to me out of the blue, would be if it makes interesting claims in topics I have worked myself. Even then, most people are usually busy enough with their own projects, students, classes, and service work, that they won't have the time to pay attention to you.

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    "it is a common experience for all (most?) mathematicians to have a "brilliant" idea, that immediatly becomes stupid when we explain it to another mathematician. We have all thrown some "earth shattering" paper to the trash when we finally realized it was worthless." QFT. The biggest problem with working in isolation is that you don't have anyone to tell you when you're going way off the rails. Unless OP is a one-in-a-billion genius the overwhelming probability is that they are most likely sitting on exactly one of these worthless bad ideas.
    – J...
    Commented Nov 8, 2021 at 19:29

So, this depends to some extent on who you are asking and what you are asking, and what your results are.

First, let's be clear that people aren't likely to to be interested if you have a claimed proof of some famous open problem, like the Riemann Hypothesis, or P != NP, or the Collatz conjecture.

Second, asking people out of the blue to collaborate is in general a not at all small ask. If you are doing so, you need to at a minimum explain in your email a) what result you think you have b) how it works and c) what part requires collaboration. Is there a specific technique or aspect that they are an expert on? For example, if someone sent me an email asking to collaborate and it was on something I don't know much about, I wouldn't respond positively.

If you have a result that doesn't need someone else, then write it up, and then talk to someone who is an expert on a closely related problem. After you have their feedback, then send it to a journal if it is good and worth looking at.

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    @Buffy Those things do circulate but the hurdle to get people to take it seriously is very high.
    – JoshuaZ
    Commented Nov 6, 2021 at 20:30
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    @Buffy, a link, please? I'd not heard anything about this, and lazy-googling is not adequate to find it. :) Commented Nov 6, 2021 at 20:35
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    Anyway, the advice to stay away from hard known problems is good advice. Such problems are much more subtle than they appear. And, of course, don't try to square the circle. I'd love a proof of the four color theorem that doesn't use a long long case breakout, though, one that isn't dependent on computers.
    – Buffy
    Commented Nov 6, 2021 at 21:34
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    @Buffy: The paper was desk-rejected, but was published by mistake due to someone clicking the wrong button. See here: twitter.com/rrwilliams/status/1417161397960646658?s=21 Commented Nov 7, 2021 at 1:57
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    @AndyPutman, Thanks for that. I got punked, I guess. I'll keep working on my proof ;-)
    – Buffy
    Commented Nov 7, 2021 at 12:08

What you are asking for here is for someone to do the job of a PhD advisor but to do it for free and without getting any credit for it at their work. If you want a PhD advisor you're going to have to apply to a PhD program.

The only likely exceptional circumstances here would be:

  1. You are a very talented high school student. People like the opportunity to work with someone stronger than their usual students, and like the idea of helping out someone like their younger self.
  2. You are extraordinarily rich and are willing to fully employ a mathematician. Given the current realities of the job market I expect you could hire a postdoc for something like $60K plus benefits as a personal PhD advisor (assuming that you make a good impression on them as someone reasonable during an interview, any whiff of crank and even that kind of money won't be enough).
  • It would be foolish of a recent PhD to allocate energy to a short-term thing like this, of course, considering that (unless, by a miracle, it does turn out to be a Big Thing), as opposed to working on more conventional ways to enhance their CV. So, ironically, if a person outside of conventional academe wants to have young people contracted to advise them, they'll probably only be able to engage people who have given up on success in academe. True, that's not the filter that we formerly pretended, but, still, ... Commented Nov 8, 2021 at 23:46
  • Probably, but it's hard to get a second postdoc, and you could finish a lot of papers if you didn't have to teach any classes... Commented Nov 9, 2021 at 0:31
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    True. An innovative way to survive? Sigh... Commented Nov 9, 2021 at 0:41

Another idea is to pay them. A university lecturer or professor normally gets paid to help students get published - and it's a full time job. Why would they do the same work for you for free? But lecturers also often offer for-a-fee tutorial services to students, mainly for those who need extra help to catch up. But a maths lecturer offering such services may be willing to tutor somebody through the process of writing up and submitting a paper instead. It might make an interesting change of pace for them. And if in the process they end up reading your work and having ideas of their own about it, that could conceivably turn into collaboration.

But don't get your hopes up too high. It's hard for outsiders to judge what's of sufficient interest to be worth publishing, and it's particularly hard for outsiders to overcome the credibility barriers raised as a result of the thousands of amateurs who seek instant fame for solving famous problems, who turn out to be fooling themselves. You should take seriously the possibility that you might end up paying someone a lot of money, only for them to tell you that the work is unpublishable, or to see your months of hard work being rejected by the journals. Such is the academic life. If that's going to upset you, don't start down this road.

As others have already said, the best way to get into mathematical academia (or any mathematical career) is by going through a postgraduate programme - this is exactly what they are for! And you are competing with everyone already on that path. Attempts to find a faster shortcut are always going to be more difficult and less reliable than following the main highway.

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    If you are a student at a university then, I'd think, that ethical rules would prohibit outside payments to any instructor at that university. It has the stink of corruption.
    – Buffy
    Commented Nov 7, 2021 at 14:33
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    @Buffy would it? If you aren't taking any classes with them I'm not sure I see any issue. And people do consulting work on the side sometimes. I would presume that the university as some ethics guidelines but that would be up to the professor to look over.
    – JoshuaZ
    Commented Nov 7, 2021 at 14:53
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    @JoshuaZ, A pretty serious violation for a professor to take "side money" from any student. It goes beyond any individual class. They can consult for others, perhaps, but not for students. A professor from a different university would be fine, though.
    – Buffy
    Commented Nov 7, 2021 at 14:56
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    @Buffy Is this the case at all universities? I taught at one university where graduate students explicitly were allowed to take money on the side for tutoring as long as the person wasn't in any of their classes. Another one had a much stricter rule. My guess is that this really does vary from school to school. (But it is possible that the one I'm thinking of was a minority. Certainly you have a lot more experience than I do about what the norm is.)
    – JoshuaZ
    Commented Nov 7, 2021 at 15:10
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    Re "A university lecturer or professor normally gets paid to help students get published": This could be read as students literally paying university lecturers or professors (probably not what you meant). Can you rephrase and/or clarify? (But without "Edit:", "Update:", or similar - the answer should appear as if it was written right now). Commented Nov 7, 2021 at 21:59

There are some good answers already. Let me add

  • There are dozens of new works appearing every day. Even if your work is correct, it's just one of n. Don't expect academics to be especially interested in your work, or take lack of interest as an especially bad sign
  • Academics get so much email. Cold emails from non-academics low priority.
  • More generally, academics are busy, and might already have many projects and collaborations on the go
  • Starting genuine, fruitful collaborations is hard. You have to have common interests, understand each other (sometimes I mean the actual language you're speaking, sometimes just the way you think about things)

As to what to do,

  • Don't get discouraged by lack of interest or difficulty beginning collaborations, though do listen to feedback.
  • Read many papers on similar topics
  • Write in a similar style, don't make bombastic claims and show knowledge of relevant literature
  • Submit to journal that papers similar to yours are published in
  • Read and take on board any referee reports you receive, or feedback from the editor

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