I'm a senior undergrad right at a small liberal arts college right now who is applying to math PhD programs in the U.S. I would like to eventually become a professor at a relatively good university that has a good environment for doing mathematics. But I'm concerned that since the schools I'm applying to are not at the very top, I am basically preventing myself from having that opportunity. For example, some professors at my college suggested that I apply to Univ. of Oregon and Univ. of Georgia, because they have strong faculty in algebra. But I looked at where past doctorate recipients from Oregon went, and they usually end up in small colleges that are pretty unknown. That is making me hesitant about applying to Oregon.
I consider myself a strong student (and so do my letter writers, and I guess that's what really matters) but I have poor math GRE scores that my professors think will severely limit my chances of getting into a program at the top 40 schools (e.g. they said I had a shot at, say, Univ. of Washington, UCLA, Chicago, and Michigan, before I got my GRE scores).
Some of my professors don't think the prestige of the grad school matters that much (i.e. as long as I write an interesting thesis and make my results known, I should be fine). But I have a feeling that my professors may be a bit out of touch, since I go to a liberal arts college, not a research university. And when I look at where past PhD students from good programs (e.g. in the Group I list in the AMS rankings but not top thirty) go to, I get a bit concerned. Maybe I should take a year off and try again next year. I was wondering if anyone had any advice.