It would be nice to have publications, but none of my current research is likely to be published, and because the research I'm doing in the future is in a similar vein as some other students I know (with the same professors), who didn't publish anything (but did give some talks at conferences), the chances of my publishing something don't seem that high. I will try, but I'm not sure how likely it will be.
I believe that, for this program at least, you apply directly for PhD. I will, of course, check.
I will definitely visit the campus and be in touch with professors I'd like to work with, once it gets closer to application time.
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At least one of the professors is quite well known in his field, though not a Nobelist or anything. They are all tenured.
The GRE is actually required, I'm not sure why I thought it wasn't. It's just the subject area that is not required, but they recommend math, CS, or physics. I would likely take math or CS.
Stanford is actually one of the few schools that weight GPA above 4.0. At least two other top 20 schools that I know of (including my own undergrad school) do this, so it's not that uncommon. That said, since everyone is coming from different undergrad universities, they might just ignore A+ in GPA, in that case my GPA would be around 3.97 when I graduate (due only to my freshman year; hopefully I will continue to maintain 4.0 from now on).
Thanks both of you for answering.