Teaching advanced courses in a specialized area of math Today mathematics (and probably also other sciences) became very specialized: every researcher works in their own narrow field which usually hardly interacts with fields of other colleagues, even on the same department.  

At the same time every researcher is often expected to accept PhD students and introduce them into his/her field. Some mathematical fields (e.g. algebraic geometry, algebraic number theory) require significant background before starting the research, and hence teaching quite a few advanced courses. 


  How this problem of regular teaching of many advanced courses on a certain  subject is resolved on your math department? Who teaches all of them? Does there always exist a group in your area on your department so that the teaching load can be shared? How it is decided who teaches what? (Please indicate your country.)
As a combinatorial algebraic geometer in a small, not highly ranked department who currently has a PhD advisee, I think I can say something about our experience.

We do not get to regularly schedule advanced courses.  With 14 grad students total, there aren't enough students to make teaching algebraic geometry viable.  We offer a 2 year sequence in general algebra, commutative algebra, and algebraic geometry on an alternate year basis, organizing the courses so that it is possible to start midway at the commutative algebra entry point.  When we have a relatively capable group of students, we just might be able to get to the definition of a scheme in the last 2 or 3 weeks of the last semester.

If we happen to have 2 or 3 graduate students at the same time who are interested in a particular topic (for example algebraic topology, or more advanced algebraic geometry, or algebraic number theory), we may organize a course for them, but faculty usually have to do this "for free" - in the sense that whomever is doing this teaching is teaching this as an extra course in addition to what he or she normally teaches.

My advisee has answered an algebraic geometry question well enough to write a dissertation.  He has not taken an algebraic geometry course.  How can he do this?  He has learned just enough of the specific bits of algebraic geometry (really just commutative algebra) he needs to tackle this specific question.  Fortunately my area attracts enough interest from combinatorialists and representation theorists that there are some reasonably accessible books and papers, but he has ended up learning what he needs to know mostly from me in our weekly meetings.  (Yes, I am disputing the claim that one needs a lot of background before starting research, even in a fairly esoteric field.)

Does he understand why the question he is working on is of interest to me and to other people I work with?  Not really.  Will he be able to find related questions and answer them in the future without my guidance?  I am skeptical he knows enough mathematics around his current problem to be able to do so, but maybe.  He is applying for teaching-oriented positions where the point of doing research is to provide research opportunities for (undergraduate) students rather than to advance knowledge, so maybe his research experience is sufficient.