When a student is undertaking a PhD in pure mathematics, is it the case that the supervisor typically chooses the topic for the student?
In short, if the advisor chooses the topic for his student in any field without taking student's opinion into consideration or a proper discussion with him about his interests – that is a very unhealthy dynamics in student-advisor relationships.
The good supervisor definitely should:
- determine the student's interests
- take into account the student's strongest/weakest skills and his goals of getting the PhD in the first place
- offer him several directions that are interesting to the advisor and might be interesting/are promising for the student
- be prepared for a reasonable flexibility within the general direction
However, it is unreasonable to expect an advisor who specializes, say, in PDE to be open for research in mathematical logic. Particularly in pure math, the initial choice of the advisor might already dictate very strongly what general direction of research you are going to work on. However, even within the general direction, there are many possible topics and approaches; thus, the list above applies anyway, in my opinion.