Question

overwhelmed not by difficulty of math but by organization of higher level math

Answer 1

> Homological Algebra which itself requires an understanding of noncommutative algebra

Uh, where did you read that?

The way you learn homological algebra is not by directly studying it for its own sake (so boring), but by taking courses in areas such as algebraic topology that actually use homological algebra to *do something*. You gradually pick up new bits and pieces of homological algebra while applying it to more interesting topics than homological algebra itself.

Answer 2

Who do you think is taking these courses? If the course is advertised as a first year Ph.D. student, then you will be fine taking it as a first year Ph.D. student. You are overthinking this by a lot, and also the prerequisites you list are mostly either not real prerequisites or just flat out wrong.

Answer 3

What about Functional Analysis?  Do you think you have the prereqs for that?

Answer 4

you can learn the necessary homological algebra during a algebraic topology course. homological algebra itself only requires an understanding of abstract algebra and being comfortable with commutative diagrams.

I think that you are overcomplicating things. just take the courses you want to take. starting with differential geometry, algebraic topology and functional analysis would probably be the best idea.

For algebraic topology you can use the book by hatcher. basic point set topology and abstract algebra should suffice

for differential geometry you can use the book by boothby or the book by Jeffrey Lee (stay away from the books by by John Lee). if you find those too difficult read introduction to manifolds by tu first.
basic point set topology and multivariable calculus/basic real analysis should suffice

for functional analysis you can use the book by conway or the book by kreyszig. here the prerequisite is a good grasp on real analysis.

Answer 5

I think you need to just talk to the instructors and ask — especially since this will help clarify some of the stuff you have in this post that’s completely wrong.

Answer 6

I feel like Hatcher is a pretty gentle introduction to Algebraic Topology that stays fairly "hands-on" with explaining concepts like homotopy or simplicial sets before moving to a more abstract setting. Understanding your preference on abstraction/algebra could help deciding on which topics to focus on.

Answer 7

>For instance Algebraic Topology at a basic understanding (Munkres) requires no more than I already have, but a level of algebraic topology that I would consider a sufficient level of understanding would be Hatcher.

It sounds cheesy but I think you need to just focus more on the journey of learning math, rather than focusing so much on *having* to master certain subjects at a high level. If you want to learn algebraic topology, and you have the prereqs to understand Munkres then just start with that and have fun with it.

It’s much better to learn *something* even if you consider it to not be a sufficient level than it is to learn nothing at all. You have to take it one step at a time with the goal of just learning something new each day. Proficiency will slowly emerge from this. Of course it’s good to have long term goals, but you have to remember that these really are *long term* goals. When learning math it’s really common that to become proficient in something you need to measure your progress in months or years rather than days or weeks.

Answer 8

You have a masters. You should be thinking about research, not learning more "topics" in the abstract. What is the exact research problem you are trying to solve, and what tools do you need to solve it?