I am a senior who is applying to PhD programs in math. I am almost done with the applications, but should I apply to more schools? Undergraduate Institution: One the top state schools (ranked top 10 in math) Overall GPA: 3.88 Math GPA: 3.95 Courses: I have taken graduate level algebra, topology, commutative algebra, real/complex analysis, functional analysis, algebraic geometry, algebraic topology, algebraic number theory, analytic number theory Math Subject GRE: 820/86% General GRE scores: 168/156/4.0 Research/Publications: -REU at William and Mary, which resulted in a paper that was recently published in Finite Fields and Their Applications - REU at Williams College (SMALL), we just recently submitted a paper for publication -Research with one of the professors in my school; we have gotten some good results but have not written a paper yet Other awards: - Putnam score of 50 (top 200) - I received the annual undergrad algebra award from our math department - IMC Third Prize - 2nd place in the annual math competition at my school I have applied to: -Harvard -Princeton -Stanford -MIT -UC Berkeley -UCLA -Columbia -UChicago -Northwestern -Brown -Wisconsin -Yale -Cornell I am a little worried now because all of the schools that I have applied to are really competitive. Should I apply to a couple of safety schools? Will I be able to transfer to one of the schools listed above if I attend some other school?