I'm a third year Physics & Mathematics student, and I want to work with a specific advisor. I have read almost all the related questions in this site and this blog post. However, almost all of them suggest the student to make a research proposal, or - sort of - a problem that s/he is planning to work in his/her Phd.
But, the thing is, I haven't had a chance to neither doing a self-study, nor taking a course in the field that I want to work on, though I'm very familiar with the basic concepts; I can easily read papers on the level of undergraduate.
In that case, how am I supposed to make -sort of - a research proposal, or suggest a problem that I can work on ?
I mean, for example, this last semester I took a graduate level Algebra I course, if I were to work on Group/ Ring theory, I could almost find fifty interesting research problems top of my head, but I'm not familiar with the actual problems in the field that I want to work on; I'm only familiar with general problems and general methods, and I don't want to look like "I'm interested with the field and to work with you, but I don't know what I'm actual interested in" in my email, so what should I supposed to do in this case ?