Friends you can count on, from Steven Strogatz’ current New York Times series “Me, Myself, and Math”.
The Toolbox, the first episode of Samuel Hansen’s 8-episode audio series featuring stories from the world of mathematics.
Terry Tao has written a probablistic heuristic justification of the abc conjecture.
This is old news, but Stanford has an archive of Knuth video lectures.
(Oh, hey, I got a job! I start tomorrow.)