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I have an idea which I'd like to work on with a professor at my university. In my initial email would it be arrogant to ask if he's interested in working with me on this particular topic (he's published some papers on closely related topics), or should I just say that I'm interested in generally doing research with him? I really do want to work on this particular idea but also don't want to come across as trying to tell him what to do.

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Make sure you include PDFs of your CV and academic transcript in your initial email. Many professors will always want to see them before deciding whether to work with someone. I know I do. – Michael Hoffman Jan 22 at 20:20
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Having recognized your handle from an earlier question, it's also worth noting that you've worked on research projects before. – Cliff AB Jan 23 at 6:46

A small point about language: in the U.S., in mathematics, it might be more tactful and less presumptious (also) to not ask about "work with me", but "advise me on" [the project you propose]. At least to my perception (=old guy) this leaves many more options for the precise relationship. E.g., experienced people can often and easily give helpful advice without really actively participating. At the same time, depending..., they might find interest in doing so. Or, at the same time, they might see that it's better to let you do it yourself... etc.

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I think that most professors like to hear about ideas that their students have to work with them.

I would simply send him an email telling him that you have an idea that you want to work on, and that you think he might be interested in.

Probably best if you talk about it face to face as most professors are too busy to read long emails.

Some thing like this:

Dear Prof X

I am contacting you as I recently had an idea for a research project on topic x.

I really enjoyed working with/learning from you on x [something he did, e.g., a class], and I am very interested in x [something he researches].

With that considered I was wondering if you would be interested in working with me on this project?

If you would like to know more, I would love to have a meeting at a time that is convenient for you.

Kindest regards,

[Your name]

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Would you recommend at some level having a research proposal of sorts outlined? At least started, so the process isn't simply "I want to work with you on this idea," but "I actually have a proposed direction for the project." Probably a no-no to send this on the first email, but in communication at some point. – CMosychuk Jan 22 at 0:30
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Good question. I think that having a literature review done would be a plus, but that any attempt to set how the project will proceed (which might be interpreted as one reason why you provided the proposal) could backfire. Additionally, the effort of writing the proposal could be futile, I notice that professors usually will take charge of undergraduate projects, both due to person preference for control and as they tend to genuinely know more about how to do the research – Peter Slattery Jan 22 at 0:34
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I would say instead of "if you are interested in working with me..." "I would need your counsel working with x". This way you enforce his/her position as a researcher – Llopis Jan 22 at 7:57
    
Yes, that is a nice suggestion. I would probably amend it again to say "I feel that I would really benefit from your help (or advice (as per paul's answer above) when working on x. – Peter Slattery Jan 22 at 10:32
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I'd have thought the only possible response to the email you suggest would be, "I've no idea if I'd be interested in working with you on your project -- you haven't even told me what it is!" There's no point inviting somebody to join a project if you've not given them any details. Better to just ask if they'd like to meet to discuss it and invite them to join you if they seem interested after that meeting. – David Richerby Jan 23 at 1:19

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