Note: This is a soft question.
It may be a bit early to be thinking about this, but I figured I'd ask now and see what responses I get.
I'm currently a high school senior, and I quite like pure math. At the moment, I'm taking Graduate Analysis at a highly ranked university out of Big Rudin. I've already taken linear algebra, differential equations, linear algebra and differential equations, real analysis out of chapters 1-8 of Baby Rudin, and topology. I'm also a course grader for Baby Rudin at the uni.
I have a few questions, and a few concerns.
Concerns:
(i) Though I've only been writing formal proofs now for less than a year, I find some of the exercises in Big Rudin to be hard to the point that I feel as though I could never prove such a thing. Some exercises seem near impossible from "down here''.
(ii) I feel as though there may be "gaps" in my mathematical education, though I'm not sure how important these are. For example, I've never written one of those super messy $\epsilon-\delta$ proofs, nor have I had to work too much with weird algebraic estimates as in some basic sequences and series. Though these "gaps" haven't seemed to pose any issues yet.
(iii) Just from studying a lot of analysis, I see truly how much information there is to be learnt. I'm concerned that when I try to learn other subjects, such as alg top, general algebra, diff top, etc, I'll slowly forget what I've learned in analysis. Like a lot of "oh yeah I forgot you could do that" when doing something simple in analysis.
Questions:
(i) Realistically, in less than a year of writing formal proofs, how able should I be in constructing more complicated proofs? How can I set goals for the future? Is there a way to quantify this?
(ii) If I maintain this sort of pace, (beginning the first year in university with algebra, functional analysis, measure theory, etc) what track does this put me on for potential graduate schools? What are students that are accepted into to 20 programs doing early on in college?
(iii) Before entering uni next fall, I have a free summer. What should I learn? What should I touch? What should I master? Naturally, algebra would be a good place to start (though I know much of it informally), but beyond this, what will give a solid foundation? Perhaps an area whose proofs are quite unique and thus give a nice broad proof-writing basis.
(iv) How should one who is just entering into the world of "real" mathematics (graduate and beyond) cope with the desire to learn all necessary graduate subjects at once? For example, I find myself wanting to learn alg top, diff geo, diff top, adv funct analysis, and operator algebras all at the same time. So is it better to get a taste of each? Or to pick them off one at a time in depth?
Finally, thank you for reading this post. Any responses are welcome and entirely appreciated. Answers can surely address my specific questions, but I've included questions that may be answered in general as to not seek individual advice.