So can anybody indicate whether it is worthwhile trying to understand what Mochizuki did?

Question

So I am looking at some math stuff and I start looking at the abc-conjecture. Naturally I run into the name Mochizuki and so start trying to see what he did. Well, he is starting look like another Galois. Frankly it was getting frustrating ... the lack of information. It seems like no one understands what he did. Well, if Mochizuki invented a new area of mathematics that solves big problems then wouldn't it be valuable to study his work? Why aren't a lot of people moving to understand what he did? There just seems to be a contradiction between his claims and the response by the mathematical community at large. The fact that it is 2014 with no new news indicates an issue. It would be nice if someone can resolve or explain this.

Answer 1

As indicated in Mochizuki's Report concerning activities devoted to the verification of IUTeich in December 2013, Go Yamashita is going to give a three-week lecture series starting in September 2014 in Kyushu University. All together it is planned that he spends 63hours, see here. Maybe his 200-300 pages detailed survey will be finished aswell at that time.

Update (5.10.2014)

Yamashita gave his first part of his 3weeks lecture series, the next one will be hold in March 2015. Interestingly, at his homepage, he has now an entry in subsections "Articles":

• A proof of abc conjecture after Mochizuki.

  In preparation.

Update (2.11.2014)

According to Mochizuki there will be a two-week workshop at RIMS from 9.03.2015-20.03.2015, called On the verification and further development of inter-universal Teichmuller theory. During that time, Yamashita will give his two-week lecture (68.h in total) lecture on Inter-universal Teichmuller theory and its Diophantine consequences.. Proceedings will be published in english in RIMS Kôkyûroku Bessatsu.

I guess after this workshop, many people will have a chance to investigate Mochizuki's work.

Update (25.12.2014)

Mochizuki uploaded a 17 pages long report on the verification-process in 2014. He's mentioning that three researchers have now read the preparation-papers as well as the full IUTT papers several times, have had dozents of seminars with him, preparing a detailed independent write-up. He is spending much space to describe the problem of motivating other people to go through the theory (due to the potential time it takes to fully grasp it), and explains some of his thoughts on how to proceed. Very interesting!

Update (03.05.2015)

A workshop on Mochizuki's work is planned by the Clay Mathematical Institute, at University of Oxford, in December 2015. Very well known participants will be there, amother others, Andrew Wiles, Peter Scholze; Ivan Fesenko (who recently wrote a survey trying to connect IUTT with well-known mathematics), Yuichiro Hoshi and Mohamed Saidi (two of the three people who studied IUTT extensively, also together with Mochizuki), Minhyong Kim (who wrote several long posts on mathoverflow on IUTT), Paul Vojta (who's conjecture seem to be [at least] discussced by Mochizuki's 4 papers), and many others. It seems like the status of IUTT will become much clearer in the end of this year.

Update (15.12.2015)

The Workshop on IUT Theory in Oxford has happened, and there are some interesting comments by participants: Felipe Voloch blogged on the different days (day 1, 2, 3, 4, 5), Brian Conrad wrote a post here (thanks Barry Smith). And Peter Woit also wrote about the conference here.

Update (17.12.2015)

There is a Nature article concering the workshop, quoting several people attending it. In the end, there is a long comment by Ivan Fesenko, the organiser of the meeting. It seems things are now going to happen.

Update (22.12.2015)

There is a long Quanta article concering the workshop, also quoting several of the attending people. Also the organizer Ivan Fesenko wrote a short report on his view.

Update (13.05.2016)

There will be a Workshop on IUTT in Kyoto, Japan, co-organized by Ivan Fesenko (University of Oxford, who organized the Oxford Workshop in December 2015) and Shinichi Mochizuki (who developed IUTT) and Yuichiro Taguchi. The participation list shows many internationals, among them Edward Frenkel (Berkley), Paul Vojta (who proposed the Vojta Conjecture for hyperbolic curves, which is claimed to be solved by Mochizuki alongside the ABC conjecture) and Vesselin Dimitrov (who is doing research on Mochizuki's work, for example here), and many others.

Update (12.07.2016)

Mochizuki released a new 115 pages survey paper on IUT.

Update (29.07.2016)

The workshop on Mochizuki's proposed proof in Kyoto, Japan seem to be much more successful than the one 9 months ago. Ivan Fesenko (organizer) writes his brief thoughts, Christelle Vincent covered 6 days from the workshop. There will be a workshop in Vermont for an introduction to concepts involved in Mochizuki’s work on the ABC conjecture, intended for non-experts. in September 2016 (see also Peter Woit's summary).

Update (01.09.2017)

Go Yamashita has published a long-awaited paper A proof of abc conjecture after Mochizuki. (see update 3 years ago). The document has 294 pages. [As a sidenote: Yamahita writes "He also sincerely thanks the executives in TOYOTA CRDL, Inc. for offering him a special position in which he can concentrate on pure math research." - which is quite impressive.]

Update (27.11.2017) According to Edward Frenkel via twitter, Mochizuki's proof is in an advanced state of peer-review. He also referes to an interview with Ivan Fesenko about further details.

Update (20.12.2017): Some critical comments appear by Peter Woit (who summarizes several recent events) and in a blog by a number theoretist (with a long comment by Terence Tao).

Update (22.12.2017): Some very concrete critics come from Peter Scholze and Brian Conrad on Corollary 3.12 in IUTT3. In addition, both say they talked with people who invested much time in IUT, but they could not solve the confusion.

Update (30.07.2018): Peter Scholze and Jacob Stix have apparently met with Mochizuki discussing their concerns. More can be read at Peter Woits blog. (Thanks Nagase)

Update (24.08.2018): The manuscript from Peter Scholze and Jocaob Stix has been published, including the statements from Mochizuki at Mochizuki's website. A very informative article about these developments can seen at Quanta, and discussions can be seen at Peter Woits blog.

Answer 2

There is one main problem with this proof being verified;

Mochizuki's argument involves a new set of ideas he has developed that he calls “Inter-Universal Teichmuller Theory” (IUTeich), they are explained in a set of four papers, which total over 500 pages.

Now in principle, you should be able to go through the papers line by line and check the arguments, making sure that no counterexample can be found for any of the steps.

The problem is that most of these steps depend on a long list of “preparatory papers”, which run to yet another set of more than 500 pages. So, one is faced with an intricate argument of over 1000 pages, involving all sorts of unfamiliar material.

I believe the only person to have completely gone through the papers is Go Yamashita, and he is now writing a 200-300 page survey, that should hopefully be more understandable.

You can read more here:

https://www.math.columbia.edu/~woit/wordpress/?p=6514