Good question. I am a tenured (associate) math professor at UGA. To be (too?) honest, I sometimes find graduate visitation days to be slightly awkward. With most of the students, they and I want to have something to talk about; for many of the students, we will have something to talk about at some future point; for a few of the students, we will become so close that it will be hard to remember these first awkward conversations.
I just remembered a visitation day where I almost sullenly plopped myself down at a barroom table next to a certain prospective student and a bunch of current students. They were more trying to have a conversation than actually having a conversation. At one point the topic turned to some elementary topology problem the prospective student had in mind for some reason. One of the current students was studying topology and she got interested and started to try to work it out. Alas I was still bored, and after a few minutes I quietly uttered a few words. The current student paused for a second, then her eyes lit up, and very soon after I got my chance to politely excuse myself and get another drink. That prospective student is now my PhD student, but the bond was not forged on that day.
But still, we should try.
1) I hear you study X. I've studied only a little bit of it so far, but I remember learning about Y. Can you tell me a little about your work?
"Can you tell me about your work?" is a very natural question that sadly can often be a showstopper. The honest answer for many professors when talking to perfectly capable students who may later turn out to be their students is often "Not very meaningfully, no." Above I tried to frame the question in a little better way. How is it better? Well for one thing, asking a professor in a department you're visiting what they do can come off as lazy: almost everyone has a wealth of information available on their webpage now. If the student really cared, wouldn't they look a little bit in advance? So the framing of the question shows that the student has looked a little bit in advance. It also clues the professor in to what might be an appropriate level at which to pitch the explanation. If X is number theory (as it is for me), then if Y is "Wilson's Theorem" I'm going to say something very different than if Y is "the Chebotarev Density Theorem".
Still it's probably best not to say too much. I remember asking one professor this when I was a prospective student and he said, "Yes, I study number theory, as do several other faculty members here. I guess what distinguishes me is that I've also become very interested in representation theory, and my work uses that to a larger extent than the others. Does that make sense to you?" The person who said this to me was Dick Gross, and yes it did make sense, and as a two sentence orientation to who he was and what he did it functioned brilliantly then and it still does, but I see now what a lot of confidence and experience he must have had to know to say so little. And he asked me if I understood, and if I hadn't he probably would have said something else: he seemed like a nice guy...and it turns out that he is. (I don't remember what I said after that. I think it would have been like me at the time to try to show some knowledge by asking a question -- that's a young math student type of thing to do. So I'll imagine that I asked "What kind of representation theory?" because I would have known that there were finite groups, Lie groups and so forth. And Gross would have replied "Algebraic groups". At that point I would have had to admit that I was totally lost...and my being lost would be of no lasting significance: suffice it to say that I have since learned about that topic!)
2) Do you do any computer work?
I think that's an increasingly important question. Nowadays a math student has to locate himself on the spectrum of more computation / less computation. If you're thinking about working with a faculty member, you definitely want to know this.
3) Are you interested in real-world applications of your work? Have you ever done any consulting or other outside work?
Again, a key question. You should already be starting to think about these sorts of questions for yourself. I, for instance, have absolutely zero contacts in industry, so I would be a terrible advisor for a student who was inclined in that direction, even if we had common mathematical interests.
4) How long have you been here?
That's a nice question for one human being to ask another. The professor will probably respond with some personal information about the town or their housing situation or their family or...something.
5) Where did you get your PhD?
You would think that would be a kind of backward-looking question to ask a professor, but actually academics ask each other that kind of thing all the time. Sad to say we really do try to have little portions of the CVs of hundreds of people memorized, even though we can look much of it up instantly if needed. Also this question subtly asks the professor to put himself in your shoes. It may also lead to some kind of insightful contrasting between the professor's PhD experiences / program and the current PhD program. On the other hand many faculty did their PhD at better places than the ones in which they are currently employed, which can be a little embarrassing. But there are various ways to spin this; maybe a temporal comparison is more interesting than a programmatic comparison. If a faculty member is old enough they can tell you that their graduate stipend was 75 cents a day plus all the potatoes they could carry, or something fun like that.
6) Do you have any PhD students? [Then ask plenty of followup questions.]
That's getting serious. You want to find out whether the professor is even on the table for being a potential advisor. And if a professor names his students, you can then go and seek out those students and talk to them about their advising experience...that can be very enlightening.
7) Are you happy here?
That's a bold question, and you might try to come at it a little obliquely, but again it's a very standard question for academics to ask each other. Currently academia has an itinerant feel to it: we are all looking elsewhere around the terrain and asking each other, "Hey, long time no see, could you show me a blade of grass over there? Hmm, that's pretty green, right? I mean, do you think? How does it compare to this blade, would you say? Greener? In what ways?"
Sometimes when you ask this question you may hear that the professor has already made or is currently making plans to leave!
8) How would you compare this program to other programs Y and Z?
Again, we're good at this, by and large. Even if we inflate our own program (it seems to be called for to do this at least a little), we are likely to be quite honest and rather knowledgeable about other programs. If you go to five different places and ask around, you're likely to get some good consensus data.
9) What advice would you give to a student who wanted to be successful in your program? Are there any specific pitfalls to be avoided?
That's an obviously good and relevant question.
Note that you can't ask "Will I be happy here?" for a simple reason: we don't know. If you're lucky, we know here and we know us. We don't know you. I think that you can in some ways get better information about the answer to this question by talking to students than to faculty, but any way you slice it, it's something you're going to have to largely figure out for your own. No one at the visitation day is going to know you like you do, after all.
Good luck.
