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?
 A: An update by Brian Conrad is here:
http://mathbabe.org/2015/12/15/notes-on-the-oxford-iut-workshop-by-brian-conrad/
A: 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.  
A: 2020 update: Quanta magazine published this article in September 2018 essentially saying that Fields medalist Peter Scholze had had doubts from the beginning, where after reading the entire work he could already say exactly where the heart of the proof was and this was exactly where he ceased to understand the argument. 
He and Jakob Stix, an expert in the relevant field of anabelian geometry, visited Mochizuki in March 2018. In the end, they both remained convinced that there is an unfillable gap in the proof, while Mochizuki keeps claiming that it is true.
Here is a report written by Scholze and Stix where they explain their point.
In April 2020, the material is finally going to be published, even though the issue was not really settled.
