This question is possibly ill-advised. (If it is not right for this site I will delete it.)
I, suddenly, have students.
It is very clear to me that there is nothing in my education that has prepared me for the task of training graduate students.
Yes, I know that graduate school is the place where one finally assumes full responsibility for one's own mathematical progress. It is also equally clear to me that there are innumerable things that an advisor might do, unwittingly, to irrevocably damage the career of their own student. This is keeping me up at night. And unlike searching for advice on, say, parenting, it appears that most people keep their opinions on the process to themselves, especially with respect to issues specific to training mathematicians. The more senior people I have approached have generally told me that "things work themselves out".
I see people I know, not so much younger than me, for whom the job market is not working itself out.
I was very lucky, and as a result I have many questions about things I didn't deal with myself. I don't know how to strike the balance between a doable research project and a significant one. I don't know how to help students move from reading background into exploring on their own. I don't know when and how much to help when they are struggling, or what to say when they become unhappy about their progress.
And I don't know where to find resources to do so. As I've said, sometimes I don't know that people take my concerns seriously… my own mentors deal with students at an n'th rate university, rather than an 8n'th.
Any direction would be appreciated.
(This question is anonymous, but not for my own sake.)
Best Answer
One important thing is to make sure your students talk enough to other mathematicians, by introducing them to people at conferences or visitors to your university, encouraging them to talk regularly with other faculty, making sure they get to know some of your friends and collaborators, trying to help them find other mentors, etc. Ideally, they should have substantive interactions with a mixture of other specialists in their area and mathematicians in other areas.
Aside from the obvious intellectual benefits (learning from many people and developing one's own identity as a researcher) and career benefits (getting good letters of recommendation), this directly addresses one of the biggest advising issues, namely the Rumsfeldian unknown unknowns. You may not know what your blind spots are as an advisor, or how to fix them even if you can identity them, but talking to other mathematicians will help your students fill in any gaps.