Even if they only call it a "visit," it is really an "interview"
I did my interview poorly, but you should not
As I explain on the page about applying to graduate school, my advisor has a personal relationship with my eventual graduate advisor, and that smoothed out the fact that (as noted elsewhere) I was also a jerk. What happened is that my eventual graduate advisor invited me out to "visit." When I got there, the "visit" came with an itinerary of all the people I would meet there. And I stupidly-yet-honestly asked "Oh, is this like an interview?" The sensible and (in retrospect) extremely kind answer was "Yes."
Dress nice
No tie is necessary unless that's your style, but business casual should be fine. Non-clinical graduate students can dress like shlubs, but interviewees cannot--or should not.
Read the department website
Don't memorize it, but be ready to recognize names and achievements that the website advertises. The department members you meet will drop names to show how awesome the department is. Note down the ones you recognize for later, and express knowing respect for the ones you recognize. Do not shrug and say "Yeah, I don't know who that is."
After people tell you what they're working on, ask them "what's next?" Chances are they have already answered that because research is always pointing into future plans, but you are there at the interview to show that you are equally curious about the future. So, ask a sensible, specific question about future directions and maybe applications. You should always be wondering about future directions, but the interview is where people get to see you do that and hear you do that.
How are students doing?
This is important because the students are best approximations of who you could be in the near future. So, ask about
how long students take to finish the degree.
where students go after they leave.
how students get funding to do their dissertation.
what health benefits students get.
cost of living in the area.
No I mean, really, are students happy? (And what is the professor support like?)
Here is where you need to keep an eye on cues from students as to whether your professor has a history as a good mentor. Questions to ask instead that will get students talking--positively or negatively.
How many students work with each professor on average in the department? (i.e., "What's the average graduate advising load?")
How many students work with the person you're applying to work for, versus with how many students work for other professors? (i.e., "How much does this professor hover around or deviate from that average load?")
How does advising work here? Like, do people get to work with other professors over the course of their time here? (It will differ by discipline how much students actually need to stay put in the same lab and only work in that lab their entire time).
And just wait to let the students talk. Why? Because students love to talk. And you will learn much about what's going on behind the website--or at least from a different perspective.
AFTER: E-mail the professor and a student PROMPTLY and then VERY LITTLE
Thank the professor who invited your visit. Short and sweet e-mail to the professor.
And ask any (few) lingering questions in an e-mail to the student you got contact information from. If no one gave you a student whose e-mail you could send questions to, then ask the professor for a student who can answer questions. BTW: you should be collecting e-mail addresses as a matter of course by now. But you should not send many questions to the student after the interview and before any decision about your application.
After the OFFER, ask students ANY/ALL (polite) questions about being a graduate student that occur to you
If the professor writes you to extending an offer to join the lab, then you should FILL an e-mail to a student with questions. Very often the professor will give you an email for a specific student--it is that student's JOB to answer whatever questions. They are part of the welcome wagon, and it is a tradition here. We know that new grad students are mostly clueless, so it's only a sign of intelligent life to have many, many questions.
If you're still reading, then that means you may really want to know about how to get in to grad school. I really can't say much more on this stupid website without inviting you to my office so we can talk about what you want to get out of graduate training.
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