Dear prospective graduate student:
1. I know you're reading my blog, and you found it from my UPenn web page, because my logs tell me that your ISP is the hotel that the prospectives are staying at. (Yes, I'm surprised too that that automatically appeared at the free site I use for such things.) Welcome.
2. Your mathematical interests will change during the first year in graduate school, because a lot of subjects "feel" different at the undergraduate level than at the graduate level, and there are some things you just don't see as an undergraduate at all. (This statement about "feeling" is incredibly difficult to make precise, but two examples are probability and number theory. Probability is usually taught in a "naive" way to undergrads and in a measure-theoretic way to grad students; number theory as taught to undergrads pretty much exclusively concerns itself with reasoning that takes place in the integers, whereas at higher levels it uses Big Fancy Algebraic Machinery. In addition, it may turn out that you think you are interested in X but in reality you had a particularly good teacher of X as an undergrad which colored your perception of that field.)
3. No matter where you go, the first year of graduate school will be painful. Maybe not so much physically painful -- but you will constantly wonder "am I the one person they admitted by mistake?" It gets better. (But bear in mind that those of us who are telling you this survived, or are about to be done surviving, the first year. The people who are currently first-years and are thinking they're going to leave the program are at this point avoiding coming to campus, so they're not here talking to you.)
4. Come to Penn! Our department is not so small that you will find no professors or other students interested in what you're interested in (with a few exceptions here and there), but not so large that you will feel like you are lost. Also, we pay well enough that you won't have to live on ramen. But you have to take a lot of classes. Maybe you like that, maybe you don't, but think about it.
(To give some context: many of our prospective graduate students are in town this weekend; they're encouraged to visit now although some have visited at other times. I said #2 through #4, in varying levels of detail, many times yesterday. I hope the advice of #2 and #3 can be useful to any of my readers who are currently attempting to choose a graduate school.)
edit, April 8, 2:29 pm: this post is also being discussed at Secret Blogging Seminar and Jordan Ellenberg's Quomodocumque. SBS in particular has some lively commenting going on.