I am an international student trying to contact a potential PhD advisor in the US. I believe I have solved an important question in my field. To contact the potential PhD advisor, I will mention that I am working on some important problems and made some progress, that I am interested in becoming their student, and then I attach my resume. I hope to impress them with my work. However, I can imagine it may backfire, maybe they think I am a crank, maybe they become interested and request my work to view it, only to find out it is wrong, and lose faith in me. What should I do?


Thank you for all the answers and comments. I decided NOT to mention my work in the email, just mentioned it in the resume with only the title. I receive positive feedback from the professors.

  • 10
    If you cannot (or do not want to) prove to them that you made progress, I would consider not mentioning it.
    – Louic
    Dec 7, 2021 at 14:57
  • 22
    I think you should try to get feedback from an expert (anyone local you can ask?) on your work, independent of your application.
    – Kimball
    Dec 7, 2021 at 14:57
  • 3
    Why not publish your solution in a peer reviewed journal. If it is accepted, and if it is important probably people will call you (metaphorically) to join their PhD program.
    – lalala
    Dec 8, 2021 at 9:50
  • 1
    @lalala, hard to get many things past an editor in a half decent journal without a track record or an institution behind you.
    – crobar
    Dec 9, 2021 at 23:52

6 Answers 6


The effect that showing your work to the professor will have is a function of several things:

  • Did you actually solve the “important question in your field”?

  • If not, did you make any meaningful progress? Or if not, do your ideas have any value at all (even if they are not novel), or are they complete junk?

  • Is there a significant gap between what you are claiming to have done and what you actually did?

In the best case scenario, you did solve the question — the professor will surely be very impressed. The problem is that this scenario is also statistically the least likely.

In the worst case scenario, you claim to have solved the question but your solution is complete nonsense. That will certainly leave a very poor impression.

Between those two extreme scenarios lies a spectrum of quite plausible situations in which you did have some interesting ideas, novel or not, publishable or not, and do not make any outlandish claims about solving famous problems but instead show yourself capable of recognizing that you cannot entirely evaluate their correctness and/or importance by yourself. As a professor, I would view such a situation somewhat positively and think that it likely says good things about you. In other words, I would be (mildly) impressed.

Basically the things to keep in mind are that these sorts of in-between situations are somewhat common, and that from the professor’s point of view, their expectations of what a student at your level can come up with through independent work are pretty low to begin with. So showing the professor anything that shows you to be a non-crackpotty, curious, independent-minded student, can be useful, and unlikely to harm your prospects. Good luck!

  • 4
    Interesting take. Do you really think there is a spectrum between solving the question and making no sense? In my experience, it's usually a dichotomy between your two extremes, with all instances being in the latter case!
    – Dilworth
    Dec 7, 2021 at 23:22
  • 1
    @Dilworth I agree with Dan - it's just all of the states in the middle are highly unstable and people either find the error in their ways or persist in heresy. More importantly practically, some people learn better by trying to make sense of things they learn and coming up with new ideas which are almost always either well-known, useless or wrong, but for not-so-trivial reasons. Having this spark is great, getting out of touch with other, especially more experienced scientists, and emotionally investing into believing you did something important without a reality check is... Not so great.
    – Lodinn
    Dec 8, 2021 at 1:04
  • 5
    @Dilworth the most common situation would be a student rediscovering some correct but known ideas. Less common but also still quite plausible would be discovering new (and correct) ideas that make a bit of progress. In both situations it’s possible for the student to also mistakenly believe they have solved the problem, although I agree that’s less likely. Anyway, my main point is that there are a variety of situations so the exact effect would depend on which one of them is happening in OP’s case.
    – Dan Romik
    Dec 8, 2021 at 1:26
  • 1
    @NoahSnyder well, sure, low quality work probably won’t impress anyone even if it’s non-crackpotty. I mean, professors’ expectations of undergrads’ independent work may be low as I said, but it’s certainly possible to fail to meet even low expectations. On the other hand, given that you became a professor, one could speculate that the seeds of your future success may already have been apparent in that mediocre undergrad paper. So I think your example can even be seen as supporting my argument (in the last paragraph of my answer) instead of rebutting it.
    – Dan Romik
    Dec 9, 2021 at 0:37
  • 1
    @NoahSnyder what do you mean by "versions" of Theorem 3 at that link: generalizations with $S_3$ replaced by other groups?
    – KCd
    Dec 9, 2021 at 10:18

Sorry to say so, but, yes, it may certainly backfire. From my own experience, if I get a PhD candidate claiming they "solved a major problem", I'm very cautious about them, and I tend not to recruit them, usually after seeing their "proofs". I definitely see it as a strong indication of being a crank, and in every case I've experienced myself the person was indeed a crank, or at least lacked the sufficient ability of judgment to evaluate basic rigorous scientific arguments.

  • 8
    This is a very good answer. To add to it - in my experience the people claiming that they did "an important discovery" were sometimes lunatics, but most often people with very small experience and ability to rigorously analyze their work. People who do not realize that their work is probably not that great when they see it themselves as revolutionary do not make good scientists. (And then there is a Ramanujan once every 100,000 cases)
    – WoJ
    Dec 8, 2021 at 14:50
  • 4
    @WoJ As evidenced by the Dunning-Kruger Effect Dec 9, 2021 at 12:02

PhDs are not about solving major problems, breakthroughs, etc. They are about learning research methods and making some real contribution to a really small sub-field. You will also need to deal with bureaucratic issues and the financial aspect.

Einstein's PhD was not about general relativity theory, special relativity theory, space-time, the nature of light, Brownian motion, or the photoelectric effect. It was not about any fancy stuff. If was "A New Determination of Molecular Dimensions."

Consider focusing on something that is a good match to you, your situation and your potential advisor/university.

  • well, that new dimension is something "everyone" knows and uses (daily) in science
    – Ben
    Dec 9, 2021 at 7:55
  • 3
    @Ben: Einstein's work was nothing about a new "dimension". It was a quite prosaic way of determining the size of molecules. And his method was soon deprecated in favor of more precise methods. Actually he was wrong by a factor of 3. Dec 9, 2021 at 18:52
  • I know, I just reformulated "molecular dimensions" ;) Well, ok, then he did quite a bad job, they should take his PhD for good.
    – Ben
    Dec 10, 2021 at 7:44

As you mentioned the key undertone that's going to happen is the professor might think you are a crank. So your first job is to convince the professor you are not a crank. The easiest and possibly best way to do this is to get an expert to vouch for you. If you are able to write something like the below, then it might become impressive:

Dear Professor [name],

I'm a prospective PhD student intending to work in [field]. I work on [topic] and have made some progress during my Honours year. My supervisor, Prof. [name2], recommended I pursue a PhD to develop these ideas further. [name2] specifically mentioned you as an appropriate supervisor. Would you ... [yada yada here's my resume blah blah]

Here Prof. [name2] is indirectly vouching for the credibility of your work. If Prof. [name] knows [name2], then it's even better.


Note that, unless your field is exceptional, for most purposes in the US, an application for a doctoral program doesn't require (or expect) that students contact a potential advisor. Likewise, in most places/fields, an individual professor has no say in who will be admitted unless they happen to be on the admissions committee. Even then, the rules require that they make decisions on the required, submitted, materials.

Instead, what you do is prepare required documents, including a Statement of Purpose, and get good, positive, letters of recommendation. The committee then chooses those that it thinks will fit well with the program and are most likely to succeed.

It is only after you arrive that you need to think about an advisor, and in most fields (some exceptions), the first task is to pass comprehensive exams. Only then is an advisor essential, but by then you have had a chance to meet some people and choose your best option.

The implication of this is that by trying to impress a professor before there is any benefit to be gained, you put yourself at risk needlessly if you are wrong. I suggest that you don't do it until you are already admitted.

True, however, that the process is a bit more compressed for those with a masters at time of admission. But, unless you know specifically otherwise for the program you are applying to, the contact is both unnecessary and unlikely to be fruitful. So, in the general case, you get a chance to sit down and have an actual conversation with any potential advisor.

For more on doctoral admissions, see the answer for the US to this question. In the general case, an email to a potential advisor might well go unanswered or you get a standard/boilerplate reply.


I will not go into details on how to get a PhD scholarship in US. All I did in my time was look universities up and go through their application process.

Related to what you're saying, it could go into your statement of research.

I would be skeptical to anyone trying to contact me and tell me they solved some important question in a field. Professors in US, and even postdocs get that kind of letters every day. Most of them are from mentally deranged people who simply have no scientific background. But if you're not one of those people, the chances that a potential supervisor would ignore your mail as scientific spam are high, precisely because of those people.

But, I remember I taught some course at a summer school this year. The way it was organized, I got to select the applicants I would work with myself. After I carefully read all the normal things (CVs, letters of recommendation, etc.) I made my final decision based on their research interests as they were reflected in their statements.

They were young people and they were gushing about how much they love their field of study and what interests them and so on, but I only looked at how well they explained what they did themselves and how. And I think I selected the best applicants.

So, if you think you solved an important problem, you should talk about it in your research statement. You should be succinct and honest. Even if your solution is not as great or as important as you originally thought, if you present your main result in a clear and logical manner, with easy to follow arguments, your application will be very good. If you already are in contact with some strong researchers, ask them for a model of a cover letter they sent when they published their work in a top journal. That's how you should present your research result in your statement.

Another thing that comes to mind is that you could have already submitted your research to a top journal, if it is as good as you say. Even if it isn't it's worth getting the opinion of an editor and of some referees. A good published work, at this stage in your career, would help your application a lot.

Finally, if you wish to contact a potential adviser, you could sent them an email saying you applied to their school's graduate program, and given your research interest, and their expertise in your field, you think you would be a good fit in their research group. You tell them what you've been working on, and attach that research statement, if they want to read it. If they are looking to hire a PhD student, they might be happy to have you, especially if your work is as high quality as you think.

  • 2
    "Most of them are from mentally deranged people": there is no evidence to support this statement, and it harmfully stigmatizes mental illness. The answer will be better if that phrase is removed. Dec 7, 2021 at 23:26
  • 5
    @GregMartin can you explain in what way you think the phrase you are referring to “stigmatizes” mental illness? Are you denying a connection between mathematical crackpots and mental illness? Or are you saying that this connection does exist but must not be discussed because it portrays mentally ill people in a bad light? Or do you simply object to the word “deranged”? Or something else?
    – Dan Romik
    Dec 8, 2021 at 19:04
  • 1
    @GregMartin It just so happens I'm mentally ill myself, and I have done that kind of stuff in during psychotic episodes. Luckily, everyone ignored me. A side effect of that experience is that I recognize people with similar symptoms, when I stumble upon them. Dec 12, 2021 at 18:57

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