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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.

Emma
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    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. – Cloudscape Sep 04 '18 at 06:36
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    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. – Brevan Ellefsen Sep 04 '18 at 06:38
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    Another thought: when approaching research (at least to start) I wouldn't recommend hoping to solve some bigger problem at the start. Perhaps you will, but its quite exceptional and rare for an undergraduate to accomplish a feat like that. Far more likely you'll spend undergraduate research doing smaller projects, gaining work experience, networking, studying in your field, etc., all of which should help to prepare you for professional research at the graduate and post-graduate level. – Brevan Ellefsen Sep 04 '18 at 06:42
  • Brevan Ellefsen, yes but my question is that I don't know how to access to "the smaller projects". – Emma Sep 04 '18 at 06:44
  • Emma, if I might ask: among these "14 undergraduate books" you have worked through quite thoroughly, which peaked your interest the most? What made you enjoy that book? What fascinated you, what prompted the most interesting questions from you, etc.? – Brevan Ellefsen Sep 04 '18 at 06:44
  • @BrevanEllefsen, The first is analytic number theory if I can do research without using advanced algebra otherwise analysis (esp complex). By using advanced algebra I mean that I learnt undergraduate algebra and a part of graduate book, but I have weakness to use it in research; and I am afraid analytic number theory would be deeply connected to advanced algebra. For example I read all the Making Transcendence Transparent book by Edward Burger and Robert and I understood all easily except last section (which I understood but with great difficultly) because of entering to advanced algebra. – Emma Sep 04 '18 at 06:53
  • @Emma I understand completely :) As an undergraduate actively researching in complex analysis and having a mind for analysis, I completely understand being shy to apply algebra to research! E.g., see here. I must admit I am thoroughly impressed that you've made such an effort to study such topics before even beginning your undergraduate. Extreme kudos! Reading through your response, it sounds to me like you've read enough to have a basic idea of what you enjoy studying which is great... – Brevan Ellefsen Sep 04 '18 at 06:55
  • ... In your case, I would perhaps recommend a paradigm shift. Instead of approaching professors asking if you can do research with them, begin actively working on a topic and be in communication with a professor studying/teaching what you enjoy. In your case, you mention enjoying Analytic Number Theory but being a bit apprehensive with the algebra involved. With this in mind, I would keep chipping away at the parts you can understand, and when you get stuck with a legitimate question don't hesitate to ask for help! This is of course easier if you are taking a class from the professor :) – Brevan Ellefsen Sep 04 '18 at 06:59
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    While I don't wish to flood this comment section (though if you have any questions for me I'm more than happy to answer!), one final note of encouragement: going into my undergraduate I hadn't read completely through a single mathematics text. I was actively studying connections between integrals involving rational functions of logarithms and their relations to dilogarithms and zeta functions, but this was purely for fun. Nevertheless, even going in more blind than you I kept chipping away and eventually got into research. It might take you some time, but with enough effort it will work out! – Brevan Ellefsen Sep 04 '18 at 07:06
  • @BrevanEllefsen, Thanks a lot :) It will take so much longer way perhaps if I wouldn't have an adviser at least for the first research... – Emma Sep 04 '18 at 07:11
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    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. – Tobias Kildetoft Sep 04 '18 at 07:14
  • I wouldn't be surprised if professors are turning you away simply due to your lack of experience. I'd echo what others have said, focus on a topic that really interests you, and when you get to something you can't understand, get a professor to help. Eventually you'll have gone to the same one enough times that they'll have no choice but to take you seriously! – MRobinson Sep 04 '18 at 07:19
  • @MRobinson, I don't want to make a bad image of people of here but with all honesty I believe that the main reason not to take a student outside of an official program is financial/promotional benefits which a professor gets from an official one; perhaps in 'advanced' countries professor would be more enthusiastic? – Emma Sep 04 '18 at 07:26
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    @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!" – MRobinson Sep 04 '18 at 07:31
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    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. – Alex R. Sep 11 '18 at 21:34

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