I was looking online for the best place to ask this question, as it seems there is no good answer anywhere. I am really sorry if this is not the right place to ask this, so if that's the case, I would appreciate if you could refer me to the right place. Here goes.

I am a junior in high school who is very passionate and curious about mathematics. I have participated in numerous math competitions (such as AMC, AIME, and USAMO), and I have completed all high school math courses up to multivariable calculus. I also developed quite a lot of experience writing rigorous mathematical proofs. In my free time, I love exploring math by coming up with problems of my own and attempting to solve them (or at least trying to make some progress). After a while, I would search it up online, and oftentimes it turns out that some mathematician has already answered it. However, the methods used to solve those problems end up being beyond the scope of my knowledge. This leads me to my question: how does a high school student like me start learning higher level mathematics, such as Analysis or Number Theory?

I know it might be a bit early for me, since I am just in high school and most of you will tell me to wait until college, but I really want to start learning now. I am deeply curious about pure mathematics, and I want to build up knowledge to a level where I can eventually get involved in mathematical research.

More specifically, here are the two questions I seek an answer for:

  • How does a high school student start learning higher level mathematics? Specifically, I am curious about topics like Analysis and Number Theory, mostly because I already have a strong foundation and experience in them (mostly from doing lots and lots of questions from math competitions). I am looking for any specific textbooks or resources that you might recommend.

  • More generally, how does one make a transition from high school mathematics to a level where one can actually get involved in mathematical research? I've been looking at famous mathematicians who achieved all these crazy things, and I've been wondering, how did they reach the level where they were at?

Sorry for the soft question, and thank you in advance for your responses.


closed as off-topic by Xander Henderson, user223391, user99914, Claude Leibovici, Morgan Rodgers May 16 '18 at 6:24

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  • $\begingroup$ Perhaps just self-study a course on Elementary Number Theory, and one on Elementary Analysis, concentrating on problems requiring proofs. Keep asking and possibly answering your own questions along the way, thus developing a research mindset. $\endgroup$ – quasi May 16 '18 at 3:46
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    $\begingroup$ "most of you will tell me to wait until college" No way, of course not. No need to wait for college. $\endgroup$ – littleO May 16 '18 at 4:17

Firstly you need strong basics to set to learn higher math. For that i would recommend "How to Prove it " by D. Vellemen. After doing this book you will have tools at your disposal to learn analysis or linear algebra. For analysis i would recommend you "Understanding Analysis " by Stephen Abbot. For Linear Algebra you can refer to "Introduction to linear algebra" by Serge Lang. Also there are video lectures on real analysis. here is Link . For learning number theory i would recommend "Elementary Number theory " by Underwood Dudley.

EDIT : My philosphical advice to OP

It seems to me from your question that you are more interested in "achievements" than mathematics. If you love something then you do not worry about achievements or prizes. You just do it. You seem to be in a hurry to get past all the math and to arrive at some "bizarre genius research world". The fun of math is in doing it now and not when you achieve it afterwards. As Watson says "Mr. Holme's work is itself reward ". Enjoy the process

Hope this helps

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    $\begingroup$ I looked over the resources that your provided, and they seem as if they're just introductory texts. The topics that they cover I already have experience with, and I need something that goes beyond that level. Also, I already have experience with writing rigorous mathematical proofs, but thank you for the suggestions. $\endgroup$ – Vlad Oleksenko May 16 '18 at 4:10
  • $\begingroup$ @VladOleksenko OK. I am beginner so i can say about these as i am using them now. Although i know advanced books on these topics also but since i havenot studied them so i cannot comment on them yet. Maybe someone with more advanced background will help you. $\endgroup$ – ReadThyOwnBook May 16 '18 at 4:34
  • $\begingroup$ My Analysis course this semester used Abbot's book and I admittingly was very much not a fan of the text. It just feels as if he lays out too much of the proof -- leaving very little to the reader to figure out on their own to learn. I'm actually the kind of person that would advocate learning Abstract Algebra first -- as that's what I did and really loved every second of it. In particular, I'd really recommend Chapter 0: Algebra by Paolo Aluffi, which was my first "advanced" mathematical text. @VladOleksenko $\endgroup$ – Andrew Tawfeek May 16 '18 at 4:53
  • $\begingroup$ @AndrewTawfeek Can you recommend some abstract algebra text for beginner and for self study ? $\endgroup$ – ReadThyOwnBook May 16 '18 at 5:22
  • $\begingroup$ @FrejaJessen The one I mentioned was my first and I used it for self-study. $\endgroup$ – Andrew Tawfeek May 16 '18 at 5:43

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