OK, so I left school (wasn't failing or anything), but I still love math and want to go on with my studies.
I want to, first and foremost, cover all the important topics a math education should cover. But, now that I get to more-or-less customize my curriculum, I've also set up personal goals of interesting topics I'd like to learn about and incorporate into my studies.
What I've seen so far:
Calculus I-III, linear algebra I & II, ODE (need to refresh on this one, can't remember much ODE).
Semester 1 of abstract algebra (groups), intro to number theory, intro to discrete maths (combinatorics, graphs, etc.), combinatorics, probability (though I really need to work on this one).
What I'm reading right now:
Second half of "Contemporary Abstract Algebra". By far my least favorite book. Messy and almost unreadable. Dummit & Foote looks a lot better, so I'll try that one next time.
"Proofs from The Book". Love it! Not a textbook, but I'm learning so much; and it's impossible to put down.
I just ordered Pugh's "Real Mathematical Analysis".
I'll start looking for a good Partial Differential Equations book soon.
Present goal:
I really want to work up to Spivak's "Physics for Mathematicians I".
From what I understand, prerequisites go up to Differential Geometry. So I figure reading his intro to DG I-III wouldn't be a bad idea.
And, if I'm not wrong, the prerequisites for his Diff. Geometry books are multivariable calculus and and differential topology.
With that in mind, is this a good sequence for completing my "undergrad" studies?:
1) Real analysis, PDE, second half of "Contemporary Abstract Algebra", Number theory
2) Complex analysis, General topology, Dummit & Foote,
3) Differential topology, Differential geometry
Is there anything missing or out of order? Is there anything that is too advanced and requires prerequisites I wasn't aware of?
Pugh is quite good.For a starter, I would recommend the lecture notes from a class by Fields Medal winner Vaughan Jones. They are fabulous.
http://sites.google.com/site/math104sp2011/lecture-notes
As far as algebra goes, I started with D&F. It's encyclopedic and I found it cumbersome to learn from. I switched to Artin's "Algebra". It is very good and a pleasure to study. I feel you can really get it from this text.
To make it a great learning experience, you can watch free videos of another math great Benedict Gross on that material.
http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra
Lastly, although it looks like you are taking on a lot, for completeness I would also suggest you consider "Linear Algebra Done Right" (Axler)