Application Process: Reach out to the prospective professor(s) to determine funding availability and if there is a place for you. If you have a strong interest and are able to conceive of a possible proposal so much the better but this amount of experience in an applicant is rare. Needless to say anyone with NSF or other independent funding is a shoe-in. As far as I know acceptance is finally determined by a vote of the faculty of the particular department. The culture of the university is heavily academically nepotistic, in that faculty tend to favor students from schools with a long history of Columbia association, usually with emphasis on association with that particular department's collaborative partner schools. Acceptance is primary driven by: undergraduate school reputation and strength of recommendations. GPA is used as a minimum bar(typically >3.5). Past experience and your willingness to participate in ongoing research (perhaps not what you'd exactly like to do) and the faculty's awareness of your interest count for a lot. Many beginning doctoral acceptees are 24-26 and have work experience/specific well-aligned research interests. A word about differences in the program from other schools: One year to study for your qualifier exam instead of two. No preassigned research rotations, go looking early for a research adviser. Focus is on getting you to the research part fast.