re: Cornell:
“There is some debate about whether the statutory colleges are “public” or “private, nonprofit” entities. Legally, they are private and nonprofit; Cornell and Alfred Universities are private, nonprofit institutions, a status which extends to all of these universities’ components, which are not separate corporations. Also, the employees of the statutory colleges, as currently affirmed by court rulings, are private, nonprofit employees. An analogy to this relationship is a private, nonprofit health agency which, under contract with a government, regularly receives government money to operate a research institute; the whole private, nonprofit agency (including the research institute) still remains a private, nonprofit entity. New York State’s Education Law also states that the statutory colleges do not operate as “state agencies.” The fact that each of the statutory colleges contains “New York State” in their official names does not alter the private nature of the statutory colleges; however, the importance of state funding is an important factor in the private vs. statutory unit relationship.”
https://en.wikipedia.org/wiki/Statutory_college