Three PhD courses on the same semester I will begin my PhD in Mathematics next month, with no scholarship, but depending on my scores I might get one later in the course of the PhD.
My question is: it is possible for me to take three PhD courses at the same time? The three available are: Smooth Manifolds, Functional Analysis and Rings and Modules. By possible I mean humanly possible to be approved with good scores.
The Rings and Modules course covers everything (homomorphisms, isomorphism theorems, etc.) up until Nakayama's Lemma.
The Smooth Manifolds one will cover tangent vectors, vector Fields, tangente bundle, differential forms, Stokes theorem, etc, and only approaches riemannian manifolds at the very end, and some notions of Lie groups.
The Functional Analysis one covers everything (Hilbert and Banach spaces, weak and weak star topology, compact operators, spectral theory), until Schauder basis and Banach algebras.
I'm currently continuing my research in Algebra which started on my masters degree, so Rings and Modules is fairly easy for me. I've already taken an introductory course in Smooth Manifolds, so I am familiar with lots of concepts, and I know some bits of Functional Analysis too (very little).
I'm planning to finish the required courses as quickly as possible, and start the research period, because I live in another city, and the research period can give me more flexibility on this (no scholarship!).
 A: Why do you need to take three phd courses on the same semester? Yes, it is humanly possible to be approved with good scores. It's also humanly possible to win a fields medal and many have done so already... humanly possible doesn't mean much - humans are exceptional (maybe you are too, but we can't know). It would be a much better idea to take at most two courses (preferably related to your future area of research!) and get A's in both of them (getting an A in a single really hard course is a lot better than just passing on three courses).
Smooth manifolds, for instance, is a subject that normally takes a year to actually learn really well from a book like Lee's. It's also knowledge that a future algebraist really doesn't need (if you were an aspiring analyst or geometer, for instance, this would change). Functional analysis can also be especially hard if you aren't very familiar with general topology and analysis on $\mathbb{R}^n$ (actually the same is even more true for smooth manifolds). I can't speak for your situation, but I - and most normal, very intelligent students - would have to dedicate like 120 hours a week to be able to handle a semester like the one you propose (assuming one hasn't deeply studied the subjects previously and is learning them for the first time in their life, which sounds like your case). I would strongly advise against this since it'll most likely lead to a lot of frustration (the fact that this is a remote course matters very little - at a PhD level you should dedicate yourself just as much - if not even more - as you would on an in-person course). Hell, even just two of these subjects would be hard ones to get A's on in the same semester.
