I cannot seem to find an appropriate verb to use for a professor when he holds an exam, quiz, etc. so that students take it and get evaluated. Can we say a professor takes an exam? I suppose this is for the student side, right? Then, what is the verb for a professor?
The general term is administer. This includes the entire process of scheduling, writing, and giving the exam.
A professor administers an exam to her students.
My professor administered an exam in our class yesterday.
Are you going to administer that exam today?
Dr. Thompson is administering an exam in Room 212; please be quiet!
Sometimes "administer" is also used to refer to the actual process of giving the exam, but technically, the correct term for this is proctor. This would describe what the teacher is doing while the students are sitting and taking the exam.
When I proctor exams, I always display a clock at the front of the room.
Although Dr. Smith was supposed to be proctoring the chemistry exam, he was surfing the web instead!
In colloquial speech, you inevitably hear give. This works fine, and everyone will know what you are saying, but it is not quite as descriptive.
I have to give an exam this afternoon to my Biology class.
Dr. Brown is giving an exam this period, so he won't be able to have lunch with us.
When the exam is given, will it include essay questions?
When will we get grades for the exam that you gave last Thursday?
There is no direct correlating verb for what a professor is doing whilst the students are taking the exam.
If the professor is overseeing the exam, making sure that the exam is being conducted properly, then he/she could be said to be invigilating at the exam; he/she is an invigilator.
She/he may be invigiliating at an exam that she/he has set and that his/her students are taking.