My understanding is that postdocs, research assistants (which I understand are master's level analogues of postdocs) and PhD students are all paid positions except for self-funded PhD which I hear is rare, but regardless, PhD students like postdocs and research assistants are expected to assist with the specific research that the professor is doing.
But, considering that a PhD studentship is the pursuit of a degree rather than the application of acquired knowledge from a degree, how much is expected of background knowledge for a PhD applicant?
There was this quora post that I can't find anymore that asked why professors took on PhD students to assist with research instead of hiring a research team or something. I was supposed to link it as an introduction.
- Update: I found the question! Why would a professor want a Ph.D student instead of a research staff? What are the advantages and disadvantages?
Anyway, my question is different from the following because I'm asking specifically for PhD students relative to postdocs or research assistants.
I'm going to use math for examples.
If I know elementary topics of complex analysis, abstract algebra, algebraic topology, algebraic geometry and differential geometry, can I apply as a PhD student of a professor who specializes in complex analysis? Complex geometry? Functional analysis? Differential geometry?