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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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    This is an excellent question to discuss with your faculty mentor. (You do have a faculty mentor, don't you?) – JeffE Aug 15 '13 at 20:59
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    Yes. And certainly I'll be discussing this with him, as well as with other people at my university whose students have got jobs. – Anonymous Aug 16 '13 at 2:38
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    Keep him busy all the time but don't drag him if he is too slow for you. The first research problem you give him should be doable (and you don't need to hide it from him). The second one can be anything depending on his success or failure on the first. If you think he may do research in the future at all, arrange that he works as a TA for some decent level class. From the job search perspective, a no teaching position is a kiss of death for one not in the top 5%, but spending all time grading business calculus assignments is even worse unless your goal in life is to become a "calculus tzar". – fedja Aug 17 '13 at 20:03
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Let him teach.

Assuming that he decides to make plans for working at a LAC or a branch campus of a state university he will need to be ready to teach the day he starts. So encourage him to take that part of his professional training seriously.

If your soon-to-be student is really going to end up in a small department a very helpful thing is that he can continue his research more or less independently. There is the internet but it is not the same as having half a dozen people working on the same topic just outside your door. This, I feel, is more important to his future than whether or not the topic he decides on has a large or small buy in cost. That being said he will want to have some facet of his research program that can be explained to a general mathematical audience, but this does not have to be all of it.

If all else fails, find a friend of yours or a not-so-recent graduate who teaches at a school like this and ask them what they want to see in a candidate. That should give you a good guide for what outcomes you should work for in your student.

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    Encourage him to give talks on his research early and often, and give lots of constructive feedback. Probably the most important part of many campus visits is the job talk. Start working now to ensure that when the time comes, he'll nail his job talk. – Dan C Aug 16 '13 at 5:42

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