Professors can research and publish freely.
Academic freedom is a foundational principle in our universities. If public universities could prohibit faculty scholarship and teaching, it would endanger the future of our nation. The inherent job duties of college professors require protection by the First Amendment. Colleges assign professors to the classes they will teach and set the curriculum for those classes, but ultimately, the professors assign reading, write and grade exams, lead daily classroom discussions, and answer students’ questions. The history of the public university itself shows that it is intended to be the “marketplace of ideas” where free thought and debate flourish.
Of course, there are nuances in the law, but generally speaking, a public university may not punish a professor for the views he expresses in a publication, nor may it punish his views in the classroom on topics germane to the curriculum.