This is a more direct, and precise question of a previous one that I have deleted.
Obligatory Information/Background
I am currently a first-year undergraduate studying at a frequented Mathematical Institute in the United States. I am well versed in "basic" mathematics, I.e linear algebra, groups, etc…, and I have recently been reading ahead, in the form of Yau’s "Lectures on Differential Geometry" and certain Mathematical Physics papers, and I decided that I want to do research in Noncommutative Geometry due to the appeal of the work/questions Connes is engaging with this very field. I also wish to "truly" read and understand the work by Yau and other differential Geometers due to their importance in Mathematical Physics. This leads me to my next problem and first question for you all.
Where does one begin?
So far, I believe I should read through the "Rudin Trifecta" which is a coined name for Rudin’s Mathematical Analysis, Real (Measures) Analysis, and Functional Analysis book. This is in order to approach Noncommutative Geometry, which largely draws ideas from measure theory.
While doing so, I wish to learn more about Algebraic Topology due to its intrinsic link to Electromagnetism and other theories such as TQFT. Considering I already know a fair bit of Abstract Algebra, I’d hope after finishing some topology I am capable of approaching Peter May’s 2-Volume Concise Algebraic Topology course.
Now, I also wish to learn Differential Geometry due to its prolific nature in math phys (Calabi-Yau, Einstein, Kahler).
As you can see I want to learn a lot, but I have limited time, so is this really a good route for myself as someone interested in Math Phys on the Geometric side of things? You can likely guess my interests based on the things I have said thus far. If someone can point out some flaws in my plan, or point me in a direction far more efficient than my naive route, that would be largely appreciated.