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I would like to work on a research project in my particular field of mathematics. While I'm familiar with the area in general, I'm having a bit of trouble choosing an inquiry narrow enough to be productive. There are lists of open questions, but open questions are generally open for a reason, and I don't have the luxury of spending a decade on a single narrow problem. (This is also the sort of thing one normally picks up in grad school, but my advisor handed me a list of problems he found in a paper somewhere and insisted that I choose among them. Also, I don't have access to any sort of mentor or other useful figure to steer me in a useful direction.)

How do I go from "I want to learn about X" or "It seems like there's a useful connection between X and Y" to some specific, publishable result?

I would like to avoid--- to the extent that one can with something as unpredictable as research--- going down an alley that turns out to be trivial or intractable.

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Short Answer Don't be too picky at first, learn about others' work; try to come up with some problem <-> solution pairs; and then you shall find your own path; when writing them down.

Longer Answer

Fun Fact: Publishable results are based on published results and some more. You need to do 'the some more' part, and know the 'published results'. So:

1. High Rank Publications and their citations: Find the best publications in the area of your work, and see how they solve the problems and how the present their results and who they cited; and read and learn them as well.

2. Narrow Down More and More When Writing Down: Ok, now after reading some papers/journals/books, you want to write something. Here we do the following:

2.1 Problem and Solution (Potential): You might ask what problem I want to work on? Well the one you know how to solve! I might go and write a code generator and say: "well, it is fast and 'efficient'". How do I know? Well I don't know! I need to check it with others. So, we go to the next phase:

2.2 Evaluation and Background Checks: You start evaluation of your work by others' approach (and write about it). Must likely your approach is not good enough, so you go back to he first step and update this one.

2.3 Introduction and Evaluation Now you know what you are talking about, you write the introduction and conclusion about it.

Then Boom! you have a paper, that could be published. Find a conference/journal and hope for the best!

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    In mathematics (the fields I am aware of, anyway), you would publish your paper in a journal, not a conference. (Well, after putting the preprint on arXiv. Usually long after that, given how much time review can take.) – tomasz Feb 25 '16 at 20:32

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