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There are a lot of ideas which I'd like to learn more about but that would take years to reach if I follow a traditional path (where "traditional path" means a kind of education where things are learned in some kind of strict order with full theory).

I'm looking for something which can present advanced math, with perhaps less formality (can meaning be conveyed in other ways?), so that I have some concrete idea of (1) what is being investigated and/or (2) what results can be achieved with these more advanced ideas. It would be great to have exercises. I have some examples of what I consider advanced:

  1. topology (algebraic?)
  2. algebraic geometry
  3. homological algebra
  4. functional analysis
  5. lie theory

Are there such resources available? Are there textbooks like this? What about for fields outside of the ones I mentioned?

(I realize this question is personal. Suppose I have a background in basic analysis, abstract algebra, and linear algebra. )

Sorry for the soft-question.

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  • $\begingroup$ 2, 3 and 4 are very technical disciplines -- lots of the celebrated results in these fields will sound extremely abstract and often unmotivated to you if you are missing out on the context of the questions that they are answering and generalizing. For 1, I have seen a book in Russian by Victor Prasolov (possibly translated?) and some popular ones in English whose authors and titles I don't remember. For 5, maybe Claudio Procesi's "Lie Groups"? (I have to admit I know a lot less about Lie groups than someone of my academic state ought to.) $\endgroup$ – darij grinberg Dec 3 '14 at 23:43
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The main reason these aren't introduced earlier is that math is very linear. You need to build up to advanced concepts, or the terminology becomes impossible. That said, there are a couple of text books I can recommend that are accessible to advanced undergraduates:

For algebraic topology, try A First Course in Algebraic Topology by Czes Kosniowski.

For algebraic geometry, I found Elementary Algebraic Geometry by Klaus Hulek very useful.

For functional analysis, Walter Rudin's Principles of Mathematical Analysis contains a chapter with a good introduction.

I'm not aware of any book on homological algebra that doesn't require some serious algebra and topology background, and it seems unlikely that there is one.

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