Apologies for the soft question. I have been puzzling over what to spend my time studying. I'm a high school senior, who only recently (during the pandemic) became interested in mathematics. I've tried to make this question both specific (so that some of my confusion may be ameliorated) yet general enough to possibly help others.
Lack of time due to schooling has kept me from spending nearly as much time doing mathematics as I would like, but so far I've learned basic undergraduate analysis from S. Abbott's Understanding Analysis, and am currently getting to the end of J. Gallian's Contemporary Abstract Algebra. I've gone through the first 150-or-so-pages of the graduate-ish Real Analysis in the Stein & Shakarchi Princeton Lectures in Analysis series to get a handle on measure theory and Lebesgue integration. I also have a non-rigorous understanding of linear algebra picked up when learning with G. Strang's textbook. My low coverage is partially exacerbated by the fact that I compel myself to complete every exercise in the textbooks I read (Gallian's 75+ exercises/chapter makes this rather difficult at times).
Long-term-goal wise I wish to be a pure mathematician for a while but likely will want to apply my knowledge to physics or computer science or the like later on.
As soon as pesky exams are over, I am itching to get back to mathematics. But I am facing a dilemma of sorts: I have so many gaps to fill, but not sure which gaps I should be filling. I am aware that the best answer is probably something along the lines of 'pick one and go with it,' but I'd really appreciate any more specific advice.
A more "for comprehensiveness" route: Learn PDEs/ODEs
My main desire, I think, is to start going through the whole Princeton Lectures in Analysis Series (in order: Fourier Analysis, Complex Analysis, Real Analysis and maybe Functional Analysis), as I found that I enjoyed learning analysis marginally more than abstract algebra. But I feel rather insecure going into these texts with absolutely zero knowledge of ordinary or partial differential equations, especially when the Fourier Analysis book starts with a discussion of the heat equation. Logically, I could spend the summer learning these things, so when I get to university, I'll be prepared to start learning analysis on a more steady foundation.
The thing of it is, I've never really done anything with differential equations, so however I learn them it'll be in a more computational setting (I mean, proofs are amazing and all, but I still have to learn how to work with differential equations in a manner more similar to a high school calculus class). I obviously can't go straight into a 'rigorous' ODE textbook like any of Vladimir Arnold's, especially with my lacking knowledge of pure linear algebra.
The 'problem' with this is that I feel as though it'd be the best use of my time to continue growing more comfortable with proof-writing, and spending time working with PDE's and ODE's isn't necessarily the best outlet for that. That brings me to my second option.
A more "for proofwriting skill" route: Number Theory, etc...
I think the most natural, given my now adequate-enough understanding of group/ring/field theory from Gallian (although I do understand this is nowhere near Dummit-and-Foote level), is to learn some number theory, which I am rather excited about. If I chose to do this, I'll likely go with A Classical Introduction to Modern Number Theory by Ireland and Rosen, as it seems to fit my purpose and prerequisite knowledge. This'll be more than enough to occupy a whole summer, especially with my 'do every exercise' credo. If I have spare time, I'll probably spend it formalizing my linear algebra knowledge with something more formal like the Linear Algebra Done Right book.
This way, I get to strengthen my 'pure math' muscles and gain mathematical maturity, which arguably is more important than learning the nevertheless essential topic of ODE's and PDE's right now.
I was wondering what pathway makes the most sense, given the limited time. Of course, I wish the summer could be endless and I could learn all the math in the world.
Thank you very much