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Here's a presentation of the mathematician the article is about. It's on the nature of "Inter-universal Teichmuller Theory"[1] which apparently was the work leading up to the proof.

[1] http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20...

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I recommend reading Martin Weissman's explanation from MathOverflow:

http://mathoverflow.net/questions/106560/what-is-the-underly...

[edit: Oops. Fixed attribution.]

That explanation was not written by Gowers at all (his name appears on it now that the answer was marked community wiki, becuase he is the most recent person to edit it, to fix a typo), and indeed although Gowers is certainly brilliant, this is far from his domain of expertise. That answer was written by Martin Weissman (username: Marty), the author information is at the bottom right corner of any answer and here says "Marty 98%".

Also, take a look at Minhyong Kim's answer.

Mochizuki has also been keeping an informational, yet succinct diary of his progress for a few years: http://www.kurims.kyoto-u.ac.jp/~motizuki/thoughts-english.h....

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