[ip26]

Another problem with overly specific interests is you back yourself into a very small arena. If the math department is accepting 1,000 students but they only have one or two professors interested in PL in the department, you are no longer competing for 1 of 1,000 slots, you are instead competing for 1 of 10 or 30 slots.

It also struck me that you said you wanted to do undergraduate research. Did you mean you wanted to work as a research assistant- or did you say you wanted to conduct your own research...? I don't think the latter really happens. I could be out of date.