So I've been researching my field of interest and I'm increasingly getting drawn to apply to a few "top" departments (in pure math) -- it seems like such a strong fit at the moment given that I've done relatively serious undergraduate research, though it was more of a "directed" study kind of thing, and my approaches to some of the problems have been fruitless, but have gotten me exposed to so much mathematics that got me excited to learn and think more about the topics I was looking at initially. I have the grades and the GRE scores (past the 80th percentile on the subject test in math, which seems like the unadvertised minimum). However, my reference letter writers don't appear to have any connections to some of these top places. From some research online, it appears that admission committees want to hear from known mathematicians -- but what if my reference letter writers aren't known in said department? Is such an application doomed from the start?
For people who have served on admission committees, what is your take on this and how would you advise an applicant like me? Unfortunately there's no time to rectify this by going to an REU or the like.
It's not necessary for your recommenders to be famous. It certainly helps a little, since it's only natural for the admissions committee to be influenced by fame, but it's not really relevant to the quality of the recommendation so most people try not to be too influenced by it. Instead, these criteria are more important:
Do your recommenders know you? I.e., can they make detailed, substantive comments, beyond just listing grades?
Do they know what's required for success at the schools you are applying to, and what the competition is like? Do they have a good basis for comparison in your case? The best case scenario is if their students regularly apply and are at least sometimes accepted; then a comparison with past students is very informative.
Does the committee understand their personalities and writing styles? Some recommenders are very optimistic and say everyone's great, while others lean towards understatement. If members of the committee have read previous letters by your recommenders, or know them personally, then it's much easier to make a reliable judgment. Otherwise, everyone will be at least a little more skeptical of the letters.
There are plenty of people who satisfy these criteria but wouldn't register as "famous" in the community at large. On the other hand, if your recommenders don't satisfy them, then it will be harder to be admitted to a top department. I wouldn't go so far as to say it's impossible, but this is worth keeping in mind when selecting letter writers.