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?
This is a short summary of the interview. The entire interview took around 1 hour. First the professor read my transcripts and found I had took a course about Fourier analysis. Hence he asked me what is the most useful technique I had learnt in that course and what were the most interesting materials of the course? Then he asked me about my understanding of graduate-level real analysis. He asked me to state Lebesgue dominated convergence theorem, and asked me why this theorem is correct. I tried to prove the theorem using Fatou's lemma and he thought it's too technical. So after several attempts under his guidance, I finally found out what he wanted is an intuitive understanding of Lebesgue integrable function (what kinds of properties that a Lebesgue integrable function has, but a riemann integral function does not necessarily have). Finally he asked me to prove some statements/ find some counterexamples related to Lebesgue differentiation theorem, which I thought were selected from PhD qualifying exam. I didn't know much about the differentiation theory at the time, so I just finished what I knew.
Basically the professor only asked me a few questions in the interview and he turned out to agree to take me as a student even though I didn't solve all the problems immediately.