The way I go about learning mathematics by my self is the following: I set up a goal, for example, the most recent one was "Complex analysis", and then I learn my way up to my goal, for example, what I did for complex analysis was: Calculus ->real analysis -> multi variable calculus -> vector calculus -> complex analysis.
Now that I've learned Complex analysis and competed my goal it's time for another and I decided that I wanted to learn differentiable manifolds and manifolds in general. My question is: What is the "path" that I should take, and what math subjects should I learn in order to be able to learn and understand differentiable manifolds in an medium / advanced level?
Best Answer
multi variable calculus -> general topology -> manifold
I suppose you have already learned linear algebra before multi variable calculus.
To comment:
I've learned geometry in only undergraduate level, so this isn't preliminary knowledge enough for advanced level.
You need not know algebras to learn fundamental properties of manifolds, tangent vectors, differential forms, and integration on differential forms. Calculus is more important than algebras to understand manifold theory. Group and module theories might be helpful to learn Lie groups and Lie algebras and vector bundles respectively, but you only need to know fundamental theorems on homomorphisms.