I was reading a GILLIGAN’S ISLAND website which says the character of the professor was only a high school professor, and serveral other sites back this up.
Now I thought the original guy was a high school teacher and when they recast the roles of the Professor, Mary Ann and Ginger, the Professor was a college teacher.
Why would a person with so many degrees teach high school? Yeah and I realize there’s no rhyme to anything on GI, but it’s fun to think about anyway 
I’m pretty sure I’ve read in another thread here that they felt the Professor looked too young to be a college prof and so made him a high-school teacher. I think he was also a Scoutmaster.
I think they changed him around several times, just as they changed (or added to) his areas of expertise. He had multiple Ph.D.'s and expertise in biology, botany, physics, engineering, literature, history, theater, and pretty much whatever else was needed that week. Science in general seemed to be his bread and butter, but I don’t recall him ever once mentioning his students or what/where he taught.
I saw Russell Johnson on a reunion show some years ago (Bob Denver was still alive) when somebody asked him some trivia about the show (I think, specifically, what the professor’s degrees were in). He was a bit gruff and said something to the effect of “I can tell you his name was Roy Hinkley but as for where he studied and what he taught, I don’t remember any of that crap if they ever even gave it.” That said, the TV movie Surviving Gilligan’s Island (a movie about the making of the series with the then survivors minus Tina Louise discussing their memories while actors playing them and the other characters reenacted some behind the scenes narratives) was, considering the source material, astonishingly good; I’d recommend it for anyone’s Netflix queue if they like pop culture.