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I have quite a few topics I'm interested in, and with many of these it may be a while before I cover them at university - I'll not be there until October, so I'd like to just get on and learn about them now as I will have a fair bit of free time soon. I find mathematics interesting in itself enough to read ahead a bit, and at the same time plan ahead how I'll get to certain topics.

I'll not be learning all of these at once, and it may not be the best idea to ask for information on all of them, I'll admit, but can anyone suggest (preferably the least number of) books relating to these topics, with any prerequisites that couldn't be covered by books for the other topics on the list?

  1. Foundations (ZFC axiomatic set theory, I so far have read P. Suppes' text)

  2. Analysis (Real and complex, to roughly a second or third year level, I've started reading Spivak's "Calculus" text, but I'd like to get even further with analysis when I'm done)

  3. Abstract Algebra (I've read W. Nicholson's "Introduction to Abstract Algebra", which goes quite far, though I'd like to read further)

  4. Linear Algebra (To the extent required for the other topics, I'm currently reading Anton's "Elementary Linear Algebra")

  5. PDE's and ODE's (Introductory, with a foundational element if possible, I'll admit I haven't read much about these)

  6. Topology (As much as possible, both point set and algebraic, I've looked into Munkres' book, which seems to be well received)

  7. Differential Geometry/Manifolds/non-Euclidean Geometry (Just a general introduction which goes quite far into these topics, I don't know much about them apart from qualitative aspects, and I'll admit I'm not entirely sure if the above so far is enough of a preparation)

  8. Commutative Algebra (Introductory book and further possibly, whatever would follow on from my Abstract Algebra book)

  9. Algebraic Geometry (Introductory book, as above)

  10. Number Theory (Algebraic and Analytic, covering such things as PNT etc. If P-adic numbers aren't too advanced, and would fall under this category I'd like to know about them, too? I currently own "A course in Number Theory" by H.E. Rose, and I'll be reading it once I've finished with Algebra and Analysis)

  11. Lie Groups/Algebras (Interested in the idea of differentiable/continuous groups. I don't know if I wouldn't know enough to attempt to learn about these, though)

Are the books I've read/am reading/have looked at good for these topics? Are there any you could suggest further for any of the topics? Have I missed any glaring prerequisites which would make it very difficult to go forward?

I know this is quite a large question, and feel free to say I shouldn't be planning so far ahead in my reading list, but if you have anything to contribute and wouldn't mind doing so could you, please? At the very least it would give a fairly comprehensive "Guide to buying books for Pure Mathematics" I'd hope. It saves me badgering you with 11 questions, too.

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  • $\begingroup$ For abstract algebra and linear algebra, I would recommend learning them from a book that combines the two, for example Artin's book. If you don't mind a dry theorem-proof style, you can also use Godement's book. I wouldn't bother reading a point-set topology book. Most introductions to analysis have enough topology for the level of analysis you're studying. For example, Rudin and Apostol's basic analysis texts do metric spaces. Rudin's more advanced Real and Complex Analysis, Lang's Real and Functional Analysis and the book by Kolmogorov and Fomin, all have enough on topological spaces. $\endgroup$ – Keith Jun 9 '15 at 16:51
  • $\begingroup$ I would say 2, 3 and 4 are the most important to start with. What you do for 1 depends on how far you want to go with it. If you just want to know what the foundations of set theory are, then there's not much more to do after Suppes. If you're interested in independence proofs, then you need to take a step back and learn some logic and model theory before going further. The others all depend logically on 2, 3 or 4 (except perhaps 6, but this is probably not true in terms of problem-solving, where it may be best to have had a basic analysis course first). $\endgroup$ – Keith Jun 9 '15 at 16:59
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I am telling my favourites-

  1. Introduction to Set Theory- Karel Hrbacek, ‎Thomas Jech
  2. Real analysis- Howie and Understanding analysis- Abbott Complex analysis- Zill & Shanahan and Complex analysis - Lang
  3. Abstract algebra- Dummit & Foote and Contemporary Abstract algebra- Gallian

  4. Linear Algebra- Friedberg, Insel and Spence & Linear Algebra- Hoffman & Kunze

  5. Differential equations- Sheldon Ross No idea about PDE, but you can check out Simmons book
  6. Topology- Munkres or Topology- Dugundji. Dont know about algebraic topology but see
  7. An Introduction to Manifolds- Loring Tu
  8. Undergraduate Commutative Algebra- Miles reid (for beginner/self study) Commutative algebra - Atiyah (Classic) Steps in Commutative Algebra-R.Y.Sharp. (I like it)
  9. Hartshorne's Algebraic Geometry

  10. A Classical Introduction to Modern Number Theory- Ireland & Rosen

  11. Introduction to Lie Algebras-Erdmann & Wildon
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  • $\begingroup$ Amazing! Thank you :) I'll wait a bit for more answers $\endgroup$ – Nethesis Jun 8 '15 at 19:06

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