I am about to officially become the Ph.D. advisor to a math grad student at a second-tier state school. (He is about to pass his comprehensive exams.)
He is strong, motivated, and talented, but not to such an extent that I wonder why he's in our program and not some top-20 program. He doesn't know precisely what he wants to work on, and will be looking to me for guidance. He hopes to get some kind of academic job afterwards, but hasn't thought about precisely what.
How can I best help him? For example, should I urge him to work on something difficult, technical, and of interest to experts -- or something less exciting to experts, but easier to explain and motivate?
Typically, although not always, our successful graduates don't get offered research postdocs, but do often go on to faculty positions at second-tier liberal arts schools or branch-campus state universities. Assuming he would be happy with such a position (which of course I will ask!), what can I do to maximize the chances for his success?
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