TITLE: What to self-study, after studying AP Calculus BC and finishing Rudin's Principles of Mathematical Analysis?
QUESTION [16 upvotes]: I am a senior in high school who has taught myself Calculus BC and I got a 5.  However, I have taken all the math I can at my school.  I also taught myself multi-variable calculus and much of differential equations, and some real analysis.
I plan on finishing my study of differential equations before school starts in early September.  I am currently working through Rudin's Principles of Mathematical Analysis and plan to finish that by the end of the first nine weeks of school.  I have not had linear algebra, but I am a fast learner.  What course progression would you recommend for the next 3 nine weeks, if I were to do a subject per nine weeks (I plan on spending a lot of time on this independent study as I am given in school time)?  What textbooks would you recommend?
P.S. I am very interested in analysis and learning more about topology, but more generally pure mathematics.

REPLY [2 votes]: I would suggest reading a set of Open Problems {the idea is not to solve them; not yet!!} but to get to start thinking about these problems in your own way. They are very motivational. The idea is to get started before you reach graduate school. Many open problems in Mathematics are easy to understand and can motivate you to read the literature. Is that not putting the carriage before the horse??? Not quiet. You need to get used to this problem solving  - develop your own method of solving these problems. There is a huge list on Wikipedia -- Mathematics Section. Google "Open Problems in Mathematics"
Professor Paul Garrity has a nice site on "Algebraic Geometry". It starts off with high school algebra and move on to Graduate Level.  http://math.lssu.edu/bsnyder/PCMI/MathFest/compilemeforamsbook.pdf ... nice for self study.