So, I did the interview. I felt like the PI seemed rather vague and tepid about any sort of next step--like having me visit campus--but by the time I got home this evening, I had an invitation. Her words were, "Thanks for your time today. I would like to invite you to visit us when works for you." Perhaps vague and tepid once again, but an invitation nonetheless.
I'm not sure what this means, exactly. I live not too far from Palo Alto, so if she is paying for the visit (not sure yet), it's not a big outlay of capital. Has anyone done one of these? Is this now a for-real interview?
For those who might care, the questions were, essentially:
What field do you intend to study?
What other schools?
Why Stanford?
Where are you aiming for, after your degree?
Incidentally, I was _very_ glad that I brushed up on actual papers--not just hers and her students', but other relevant and current papers--and had the notes in front of me. Thanks for that advice. I think that really bailed me out of some other, not-so-hot answers... I know they weren't so hot, because after the fact, she told me what she hoped _not_ to hear from applicants, and I said some of those things.