If so, then they can enforce that trademark if they feel that your name can cause confusion and dilute or harm their own trade mark.
Fortunately University of Columbia is not the same mark as Columbia University.
Tom, you are describing a trade school or a career college, not a university. There is a difference.
There will be various research and lecture functions to the school as well.
Zetetic articles and papers can be published for and maintained by the University of Columbia. For example, some of our members are in the process of writing books and papers on various subjects of interest. Their works can be maintained and publicly accessible by the University of Columbia.
Also, if a member of this website were to ever lecture on Zetecism or Flat Earth Theory at another college, as Samuel Birley Rowbotham and Charles K Johnson did in their lifetimes, he or she can do so as a "Visiting lecturer from the University of Columbia".
I feel that the undergraduate school of UoC should operate as a career school as I've described, with videos lectures and assessments for specific career types. The Undergraduate School will be the main funding source for UoC, and the most popular, allowing the school to be competitive in the marketplace, providing short and inexpensive routes to a college degree.
The Graduate School, however, will be primarily research based, and quite different than instruction at the undergraduate level. At the graduate level it is assumed that the student is a maser of his field and must now contribute something substantial to it. There will be no courses at the graduate level. At the graduate level the student spends his time working on a major research dissertation, which will be his or her primarily project for the school.
Funding for the school will not be an issue. Both the Undergraduate and Graduate schools of UoC can operate on a shoe string budget with minimal staff. At the undergraduate level the student is basically teaching themselves with videos their adviser selects, and at the graduate level the student keeps themselves busy writing their dissertation. UoC will not need to invest heavily in a campus or staff. Also, as a Non-Profit, UoC will not be burdened with taxes.
Eventually, as the school grows it could even branch out into operating real physical classes with paid instructors. Even Harvard started small, after all.