What is the status of the purported proof of the ABC conjecture?

Back in August 2012, Japanese mathematician Shinichi Mochizuki announced a proof of the abc conjecture using Inter-universal Teichmüller Theory. What has been the status of his proof? Has there been any progress made in verifying it?

Solution 1:

An update by Brian Conrad is here:

http://mathbabe.org/2015/12/15/notes-on-the-oxford-iut-workshop-by-brian-conrad/

Solution 2:

There is no real update here. At the moment no experts outside a somewhat close circle have made any positive claims regarding the arguments, and if anything they are (on average) somewhat doubtful.

There is a basic difficulty that no-one knows where to really begin in order to understand what is going on in the argument, and Mochizuki more-or-less refuses to explain how the argument goes other than to say "read the papers".

The situation is peculiar; indeed, it may well be unprecedented. One possible outcome is that a group of researchers around Mochizuki accept his argument and techniques, and publish results building on it (say in relatively minor journals, where the fact that many people don't accept the results won't interfere with refereeing/publication), while the majority of researchers remain agnostic or skeptical. This would be pretty unfortunate!

Go Yamashita has been promising a write-up for a long time; perhaps when that appears the situation will become clearer.