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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?

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@fakaffTo address your question of order, I would suggest, having done it myself: –  Andrew Jan 23 '12 at 17:50
    
Reading about what you are reading/read inspires me! :) –  zerosofthezeta Jan 28 at 4:09
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1 Answer

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)

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Thanks for the recommendations. To clarify, I won't be taking all of this at once. I just want to map out a curriculum that would more or less cover everything an undergrad education would cover. –  iDontKnowBetter Jan 23 '12 at 3:41
    
@fakaffWith regard to your question about the order, I would suggest, being in the midst of it myself: Vaughan Jones's notes (elegant and self-contained foundation for real analysis), then Pugh Ch 1-4 to have a stronger real analysis basis, then at least the first eight chapters of Axler (again a good foundation for algebra), then on to Artin + Gross's lectures - math heaven. All of these are very well-conceived pedagogical treatments of the topics. After these you will have a good sense of where you stand. If you are inclined to look at complex analysis, I would suggest Stein - very clear. –  Andrew Jan 23 '12 at 18:01
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