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When I was in my high school I studied 14 undergraduate books with almost all their exercises. I am about to begin my undergraduate in mathematics this week. I love to do research in mathematics. I met a few professors during the last few months and asked them if they can accept me as their 'unofficial' research student but they all refused even if after I become officially an undergraduate student but without taking "research for undergraduates" course or I have be their graduate students.

Trying to do research on my own, I found a book that includes many unsolved problems. My question is that if I choose to do research like most students, that is adding knowledge to mathematics by expanding it gradually and in smaller steps, I can't because I don't have a adviser to know the frontiers and if I want to be an independent researcher I just know the problems that are famous to be impossible to solve!

How can I take a win-win with both; that is how to find 'smaller' unsolved problems like the problems students publish papers on, as an independent researcher when nobody willing to share them?

Also I found out that if learn mathematics along doing research I memorize the materials easily after analyzing them. That's a good side-effect of research compared to only studying.

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    $\begingroup$ You have to find a specialized area. These will however require some in-depth knowledge. Probably you should pick up a book in a specialized area and work through it, and then find some papers in that area. In the latter, problems will become clear automatically. $\endgroup$ – AlgebraicsAnonymous Sep 4 '18 at 6:36
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    $\begingroup$ Welcome to MSE. Tough question that many exuberant undergraduates encounter. Kudos to you to reaching out to professors already, that's a big step. I recommend you keep trying. That being said, I would really focus yourself on specializing rather than going broad if your goal is to research ASAP. Of course, knowing what you enjoy doing and would thus enjoy specializing in is a tough problem, which ironically requires a bit of breadth! As such, my recommendation would be to keep studying what you love and keep getting experience for the time being. $\endgroup$ – Brevan Ellefsen Sep 4 '18 at 6:38
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    $\begingroup$ Expanding on the comments of @BrevanEllefsen I would also encourage you to ask any questions arising from your attempts here or in chat if you feel they may get into more chatty territory (we are generally very friendly in chat and always happy to welcome new people). I would also suggest to not focus too much on producing new math for now and just on learning by doing stuff that is new to you since that has all the benefits in the learning but doesn't involve the tedious checking of whether something is new. $\endgroup$ – Tobias Kildetoft Sep 4 '18 at 7:14
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    $\begingroup$ @Emma That's always a possibility! I'm trying to be a bit less skeptical for a change! But it is definitely true that professors will rather help someone who can show a load of work backing them up, and have a smaller problem, than someone just asking "help!" $\endgroup$ – MRobinson Sep 4 '18 at 7:31
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    $\begingroup$ I would actually advise to not rely on open problems from books. While they may be intellectually stimulating, I think that there's a deceptive element of difficulty here. By partnering with an advisor, you're not only gaining a vast well of experience to help you, but also experience on what problems not to work on, because most open problems tend to be much more difficult than they look, and potentially less interesting when contrasted with current trends in mathematics. $\endgroup$ – Alex R. Sep 11 '18 at 21:34

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