duck_syndrome wrote:As opposed to say going to an Ivy or a comparable top 20 institution
My advisor told me this "School's like Harvard, Stanford and MIT only accept the top 2 students from each of the top 50 undergrads in the world. In order for someone from a school like UIUC or UIC (we're sister schools + I go to UIC) to even have their application looked at, that student must have letters of recommendation from big professors that say 'This student is the best student I have seen in my decades of being at this school'. They also have set criterion like if Peter May (Algebraic Topologist from UChicago) says something very positive, then that's guaranteed success. Beside those three, every other program is something you have a shot at."
Something I recommend is emailing a professor and asking him to serve as your undergraduate mentor. Develop a connection with three professors because you need three very strong letters of recommendation if you want a good shot at a top 20 PhD program.
I should point out that not all Ivy league schools have top 20 PhD programs. In fact, most of the elite schools have an elite status because of the work their professors did 30 years ago. Many of these professors are quite old and are barely able to keep up with modern mathematics.