I recently had the great pleasure of interviewing Yasha Berchenko-Kogan, an MIT graduate student in mathematics who became notorious on the Internet for his answer to a question on Quora about what mathematics graduate students do all day.
Yasha and I first met at the MIT Math recital, an annual music concert at which every participant is affiliated with the Math department (Yasha performed a piece on the piano). After this initial meeting we decided to find a time to talk more in detail.
In the interview we talked about a number of things, but the primary theme was explaining mathematical concepts to ourselves and others. A number of new insights came out of the interview, and I hope you will enjoy it as much as I did. The transcript is here, and this is the article Yasha mentioned towards the end of the interview.