Lets say you had an undergraduate who wanted to do some advanced work and some research, possibly for a thesis, or things like that. 

There are two slightly more specific groups of questions I have about this process:

* How would you go about choosing a problem? Specifically, should the student work on open problems or work through existing proofs? Are there lists of problems at that might be fruitful and approachable?
* What sort of guidance would you provide them? With what frequency would you meet? Would your meetings be closer to teaching or guiding them along on their own?