<p>The bad news is that math seems to be rather prestige-oriented. For example, 18 of the 21 American admits to the pure and applied math PhD programs at MIT this year got their undergraduate degree from a top 10 math department. (Note that I am referring to graduate math rankings, not general undergraduate rankings.) The three outliers were from Duke, the University of Illinois at Urbana Champaign, and a liberal arts college that shall remain unnamed. </p>
<p>The good news is that it’s not impossible. I graduated from a small liberal arts college and - surprisingly enough - got into every single math PhD program I applied to. What tipped the scale in my favor (according to a faculty member at one of the graduate programs I applied to) were two year’s worth of graduate courses that I had taken off campus, and a letter of recommendation from an Ivy League professor who compared me very favorably to the undergraduate and graduate students at his own university. </p>
<p>Whereever you go, if your goal is to get a PhD in math, make sure to take a number of graduate courses as an undergraduate. They are basically an admissions requirement to the top 10 math programs. </p>
<p>As far as Vanderbilt vs Ohio State is concerned, Ohio State has the stronger math department, so you might be better off there.</p>