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I wanted to know are there any problems in Functional Analysis (FA) that can possibly be successfully tackled by someone like me who does not have any expertise in this area but is only familiar with a few basic topics that you would find in most undergraduate level courses?

I wanted to mention that I did look around the web before posting here. There doesn't seem to be much left that an undergrad can do in this area (or almost any other area), but I have seen sometimes papers by other researchers who in the end of their papers mention how their work can be used to do something (usually these are concrete applications or suggestions to work on specific examples), but the author didn't find the time or hadn't the resources to carry out the work and it's left to the interested reader. I wanted someone to help me find these kind of problems that would be easy to work on if I give it some time.

My goal is to write a research paper and get it published in a suitable journal. I'm out of school at the moment and would like to get admission into a good PhD program. It is very hard for someone like me to get the attention of a professor to take me as a doctoral student without having proven first that I am motivated to and can do the work in FA.

Thank you for your time and help.

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    $\begingroup$ In general, it's hard to find good research problems for undergraduates. You just don't know enough at that stage. It might be possible in more "elementary" subjects like combinatorics, but I think you are unlikely to find something in functional analysis that does what you want. $\endgroup$
    – Nik Weaver
    Commented Jan 18, 2020 at 20:45
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    $\begingroup$ "these kind of problems that would be easy to work on if I give it some time" - this is not going to be a way "to get admission into a good PhD program" and won't be a way to succeed on such a program if one gets in. $\endgroup$
    – Yemon Choi
    Commented Jan 18, 2020 at 21:12
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    $\begingroup$ To be more constructive: it's unusual to have publications as an undergrad. Math departments are looking for people who have the potential to do research. This means having a good undergraduate record and (in the US) good GRE scores. If you don't have this your best bet is probably to apply to a master's program somewhere, do well, and use that as a springboard into a PhD program. $\endgroup$
    – Nik Weaver
    Commented Jan 18, 2020 at 21:50
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    $\begingroup$ I'm voting to close this question because it seems to me to be based on a misreading of how one can or should get into a PhD program, and any problems listed here are unlikely to yield the desired results for the OP $\endgroup$
    – Yemon Choi
    Commented Jan 19, 2020 at 2:53
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    $\begingroup$ Despite my comments above: good luck getting into a PhD program, perhaps by the route(s) suggested by @NikWeaver $\endgroup$
    – Yemon Choi
    Commented Jan 19, 2020 at 2:54

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Since your aim is to show your abilities and get the attention of a professor that would take you in a PhD program, the best and most natural thing would be, getting the problems from the professors themselves. Read their papers, find their open problems that you may like, and then write to them (with discretion) for clarifications or for suggestions of "easy cases" to start with. After all, this is how great Sofja Kovalevski got the attention of Weierstrass: solving his test problems in a week. Good luck!

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  • $\begingroup$ Thank you for your answer. I have found that when I email professors whom I would love to work with, most of the time they don't want to take on new students, already have too many students to supervise, or don't have funding to take on new students (these are the most common responses to my emails and I'm beginning to suspect that it's usually just the professor's polite way of saying I lack the subject knowledge required to carry out research). I might spend an awful amount of energy reading their papers and familiarizing myself with the work they're doing only to be rejected by them. $\endgroup$
    – J. Doe
    Commented Jan 19, 2020 at 6:09
  • $\begingroup$ @ChristianRemling You are right, of course this is not the usual route; the OP explicitly says they would prefer not to follow the usual route (otherwise I guess they would have followed it, without need of asking) $\endgroup$
    – Pietro Majer
    Commented Jan 19, 2020 at 18:41

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