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:
- topology (algebraic?)
- algebraic geometry
- homological algebra
- functional analysis
- 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.