You don't want to leave a nice bike parked outside class in the sun and rain every day. I commuted on a road bike for part of undergrad up north and through grad school in the south, but the exposure was tough on it, and since then I've replaced most of the components. Up north you'll also have salt to accelerate the corrosion, and road tires will be slippery on ice and snow. Somebody tried to cut the cable lock on my road bike at the bike rack in front of the lab one evening, but failed. I switched to a U-lock and added locking wheel skewers.

A used rigid-fork MTB or hybrid would work well enough for a 3-mile commute. You can put slick tires on it for most of the year, and change to knobbies for the winter.