I am sure that I will be finding out first hand as I am entering a PhD program, but I will ask my question anyway. Say, for instance, a student has worked through the majority of a textbook like Dummit and Foote. How far removed is the aforementioned student from current research in Algebra? In the same vein, how far removed from finding a research topic in Analysis is the student who has worked through the majority of a book like, say, Royden?
I know my question is difficult to answer for a number of reason. However, I am sure that you get the "gist" of what I am asking. Thank you for you time and input!