NOVA's The Proof: Presenting Proof, Success and Discoveries in Mathematics

To help you identify the main mathematicians in the video: Andrew Wiles: Appears first;     John Conway: Escher shirt;     Peter Sarnak: Maroon polo with grey collar and later a green polo;     Nick Katz: Mustache, light blue button down / helped review the proof;     Ken Ribet: Peach sweater with blue collar / cafe / Epsilon conjecture;     John Coates: Suit jacket and tie / Wiles' adviser;     Barry Mazur: Patterned dark button down;     Goro Shimura: White button down / Taniyama-Shimura

What are the analogies or metaphors that people in the video present to describe the process of doing mathematics and/or mathematical research?

How do people in the video describe what a proof is?

What practices do mathematicians employ in order to encourage the flashes of insight they need for success in research?

Some people view mathematics as a solitary profession. What community and social aspects are portrayed?

Global Perspectives: What countries from abroad play a part?

Algebra plays an important part in the video. What are the additional fields of mathematics that relate to high school or college?

Comment on the general strengths and weaknesses of how the mathematicians presented mathematics, especially in reference to Paul Hamos' ideas of "How to talk mathematics."