As a student at Columbia, CUNY, or NYU, you may take courses with any of the faculty in the NYAARC consortium, but your PhD advisor should be at your home institution. So while you may apply to pursue doctoral studies at any or all of the NYAARC schools, it is best to enroll in the program where your anticipated advisor will be.
For more information about applying to the participating programs:
Columbia PhD Track in Archaeological Anthropology