ABC proof

Mathematicians finally starting to understand epic ABC proof

By Jacob Aron

It has taken nearly four years, but mathematicians are finally starting to comprehend
a mammoth proof that could revolutionise our understanding of the deep nature of numbers.

The 500-page proof was published online by Shinichi Mochizuki of Kyoto University,
Japan in 2012 and offers a solution to a longstanding problem known as the ABC conjecture,
which explores the fundamental relationships between numbers, addition and multiplication
 beginning with the simple equation a + b = c.

Mathematicians were excited by the proof but struggled to get to grips with Mochizuki’s
“Inter-universal Teichmüller Theory” (IUT), an entirely new realm of mathematics
 he had developed over decades in order to solve the problem. A meeting held last year at
the University of Oxford, UK with the aim of studying IUT ended in failure, in part because
Mochizuki doesn’t want to streamline his work to make it easier to comprehend, and because
of a culture clash between Japanese and western ways of studying mathematics.

Now a second meeting, held last month at his home ground in Kyoto, has proved more successful.
“It definitely went better than expected,” says Ivan Fesenko of the University of Nottingham, UK,
who helped organise the meeting.

The breakthrough seems to have come from Mochizuki explaining his theory in person.
He refuses to travel abroad, only speaking via Skype at the Oxford meeting, which had made
it harder for mathematicians outside Japan to get to grips with his work. “It was the key part
of the meeting,” says Fesenko. “He was climbing the summit of his theory, and pulling
other participants with him, holding their hands.”

Glimmer of understanding

At least 10 people now understand the theory in detail, says Fesenko,
and the IUT papers have almost passed peer review so should be officially published 
in a journal in the next year or so. That will likely change the attitude of people 
who have previously been hostile towards Mochizuki’s work, says Fesenko. 
“Mathematicians are very conservative people, and they follow the traditions. 
When papers are published, that’s it.”

“There are definitely people who understand various crucial parts of the IUT,”
 says Jeffrey Lagarias of the University of Michigan, who attended the Kyoto meeting,
but was not able to absorb the entire theory in one go. “More people outside Japan have
 incentive to work to understand IUT as it is presented, all 500 pages of it, making use
of new materials at the various conferences.”

But many are still not willing devote the time Mochizuki demands to understand his work.
“The experts are still on the fence,” says Lagarias.
 “They are waiting for someone else to read the proof and asking why it cannot be made easier
to understand.”

It is likely that the IUT papers will be published in a Japanese journal, says Fesenko,
as Mochizuki’s previous work has been. That may affect its reception by the wider community.
“Certainly which journal they are published in will have something
to do with how the math community reacts,” says Lagarias.

The glimmer of understanding that has started to emerge is well worth the effort, says Fesenko.
“I expect that at least 100 of the most important open problems in number theory will
 be solved using Mochizuki’s theory and further development.”

But it will likely be many decades before the full impact of Mochizuki’s work on number theory
 can be felt. “The magnitude of the number of new structures and ideas in IUT will take years for
 the math community to absorb,” says Lagarias.