Interview for pure math graduate program

I will have an interview for my graduate school application in the coming days. I don't have much research experience but I am taking a few graduate courses which are related with the potential research field, so I think the main part of the interview will be on my coursework. The interviewer is my potential supervisor whose research field is theoretical PDE and applied analysis. The professor is very famous and has a real authority in this research area (at least in my region).

I have talked with the professor before, and he said he will ask me to talk about my "understanding of mathematics" in the interview. To be honest, I don't quite understand what the professor mean. What are the questions a professor will ask in an interview for a pure math program? Is it common to ask me to solve some exercises immediately? I think comparing with solving a specific exercise, "understanding of math" is more high-level, for example, the development of real analysis. Could you give me an example or share me your interview experience?