crispyk

Ok, I'll bite. I'm also applying to math PhD programs. Our stats are similar but you have slightly better GRE scores and come from a better college (luckily for me there is only one school we are both applying to). I probably do not know much more about this stuff than you and your best bet would be to ask a professor at Chicago, but it is nice to have an outsider evaluate your profile. I would not hold your breath for Princeton, Harvard, MIT, or Stanford. I don't think anyone should: they all accept very few people (keep in mind that not only is the incoming class small but they do not have to admit many since most people who get accepted there end up going) and there are tons of students with subject GREs in the 90%ile, excellent GPAs, excellent letters, and publications. With that said, I think you have an ok chance at these schools. Your profile is probably a little below average for the typical admitted student at these schools (I'm guessing most have subject GREs in the mid 90s and a few publications). I'd be surprised if you didn't get into one of Columbia, Berkeley, or Cornell. Out of those three I think you're most likely to get into Cornell. I heard that Berkeley admitted too big a class two years ago and as a result admitted fewer students than usual last year. I am not sure what the size of their admitted class will be this year. I think you're all but a sure thing for Michigan. I won't comment on UCLA since I have never looked much into their program. I do not think your lack of publications would hurt you since you did do three REUs at a very good math department. I also do not think that your flirtation with a math/CS double major would hurt you. However, I do not think it will help you much unless you mentioned that you are interested in a research area that utilizes a lot of programming (computational algebraic geometry?). Lastly, I do not think your younger than average age will have any effect. I think someone would only be considered too young if they were younger than 18. Good luck!