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This may not make a lot of sense but:

I'm interested in studying more about topological manifolds, smooth manifolds, and maybe some algebraic topology later etc. but I'm not sure if I need to have a good prior understanding of classical geometry. My geometry sucks, period, and I know that modern geometry and topology provides us a modern way to approach problems in geometry; however, if I don't have a very good understanding of classical geometries (classification 2-d surfaces etc. and what not), I'm not sure if I'd be able to either appreciate or dig deeper into the subjects. For example, in analysis, we don't study functional analysis before linear algebra, and we don't study linear algebra before Euclidean geometry because each subject builds on the previous one.

Am I over complicating this issue? I need some suggestions on how to effectively dig deeper into these subjects.

user82261
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  • I would like to recommend Fulton's Algebraic Topology: A First Course which covers algebraic topology on planes and surfaces, a gem of "low-dimensional" introduction to algebraic topology. A second look at Miles Reid's Undergraduate Algebraic Geometry, which is available at his homepage. – Yai0Phah Jan 18 '20 at 01:04

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You're overthinking it a little, IMO. You really should just pick up a good book like Lee's Intro to Smooth Manifolds and see if you can follow along. Oftentimes the classical theories are presenting specific examples and cases of the more general theory. In my experience, studying a general theory can give me a foothold for understanding something "more elementary" precisely because it provides a broader context in which special cases can be understood.

Now, all of this goes with the assumption that you've studied basic courses like analysis, linear algebra, and some group theory. If you haven't studied those subjects, then you really should gain familiarity with them before studying differential geometry. However if your concern is more regarding, as you said, "not knowing the classification of 2-surfaces," then you should really just jump in and get your feet wet.

Mnifldz
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  • Yes, I'm trying to work through Lee's Topological Manifolds these days. I guess I'll keep working on it, and jump to his other book for the course I'll be taking next semester. I suppose my worry is that I don't even have a very good appreciation of the classical problems in geometry etc. that ultimately led to the creation of these modern subjects. Meh, I guess I'll just dig into these books. – user82261 Nov 11 '19 at 08:49
  • @user82261 If I may offer a gentle suggestion.... rather than judging your progress against knowing the classical problems, it might be worth asking what are the problems you want to solve and whether you're taking the right steps to that end goal. – Mnifldz Nov 11 '19 at 17:45
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    Yes, I understand your concerns. I'm interested in applied probability and applied topology, and I'm currently completing my master's in mathematics. I'm trying to use as much time as possible this year to learn new mathematics before settling in a different program that can better serve my applied interests. Yes, I know I may be taking a longer route that may be recommended in my case, but I'm trying to shore up on my math background while I'm getting this master's degree. – user82261 Nov 11 '19 at 19:50
  • @user82261 Persistent homology type stuff? Very cool, good luck! – Mnifldz Nov 11 '19 at 21:05