It is well-known that Research Experience for Undergraduates programs are
harder to get into than graduate school, but they are well-coveted, both
because they are well-funded programs which are invaluable for graduate school
admissions, and because they provide students with exposure to mathematics
far outside the scope of the undergraduate programs.
As a rising junior, I was admitted to 3 out of the 9 REU programs I applied to. Given that said REUs often admitted fewer than 1 in 50 applicants, and that they are usually meant for rising seniors, I figure I must have done something right.
Fortunately I have been able to speak to the program director of the San Diego State University REU about what makes a good applicant. According to him, there are only four things prescribed on the NSF grant, though I'm not sure if this generalizes to REUs in general. Also notably missing from this list is whether the applicant fits the program and its research project. So, while the reader should take the following list with a grain of salt, I hope that it will be useful to young applicants nonetheless.
passionateabout mathematics, so passion counts for very little; more important is that the applicant is able to persuade the REU that they have studied sufficiently advanced classes that they will not wash out of the REU halfway through, and that they are comfortable with struggle. For example, I wrote the following passage on each of my applications:
[F]or instance, as part of a project for my analysis class, I proved that for each uncountable Polish space P and each countable ordinal \alpha, there exists a function on P whose Baire rank is \alpha. This originated from a problem posed by my professor: How does one generalize the notion of a Baire rank from the integers to the ordinals? In order to solve this problem I needed to learn basic descriptive set theory and measure theory.In hindsight, it would have been wise to also mention why I even attemped this problem in the first place: I had just bombed the midterm and needed to redeem myself: this would have highlighted that I am comfortable recovering from major setbacks, as so often occur in academia.
lukewarm, the program director would just throw out the application without reading it. To ensure strong letters, it is often recommended to go to office hours frequently and to ask the recommender if they could write a
strongletter, so that they could refuse if necessary. This is probably the most difficult part of the application for a lower-division student at UC Berkeley or a similarly large school, because of how little they interact with faculty, and I am not sure it is worth one's time to apply as a freshman from Berkeley.
If anyone has any other tips, I'd be glad to hear them. Good luck!