You know, I’m something of a scientist myself.
Peter is first introduced to Norman Osborn, father of his friend Harry, just outside Columbia University’s Science Department, where the students have a tour of the new genetic research facility.
Columbia University plays itself, specifically the Low Memorial Library, at least for the exterior shots.
For interiors of the research location, and the spot where Peter is bit by the radioactive spider, the production used the interior of the Natural History Museum of Los Angeles County.
In the sequel, the campus continues to make appearances as itself, with Peter running into his professor Dr. Conners (who chastises Peter for the low-effort he’s putting into his class), making a call to MJ apologizing for missing her play, the exterior of the Student Health Services building (actually Havemeyer Plaza), and Peter exiting a class outside Hamilton Hall.
Peter is shown not paying attention to Dr Conner’s lecture in a quick classroom sequence that was presumably filmed in one of the classrooms on campus.
The only shots of Columbia University in the three-quel are these classroom shots, which appear to be a different lecture hall than was seen in the second film. The assumption is that these were also filmed at the University.
(Spider-Man, 2002; Spider-Man 2, 2004; Spider-Man 3, 2007)