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Japan

Kyoto University is one of Japan and Asia's premier research institutions, founded in 1897 and responsible for producing numerous Nobel laureates and winners of other prestigious international prizes. A broad curriculum across the arts and sciences at both und... more

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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.
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.

bit-player.org

Back in the summer of 2012 Shinichi Mochizuki of Kyoto University released four long papers that claim to resolve an important problem in number theory called the abc conjecture. (I'm not going to try to explain the conjecture here; I did so in an earlier post.) More than three years later, no one in the mathematical community has been able to understand Mochizuki's work well enough to verify that it is indeed a proof.

In terms of expository style, Mochizuki and Harron stand at opposite poles-a fact noted by at least two bloggers.
Both of the posts by Evelyn Lamb and by Anna Haensch mentioned above in connection with Mochizuki also discuss Harron's work.

www.wired.com

Caption: Go Yamashita lecturing on the work of Shinichi Mochizuki.

Philipp Ammon for Quanta Magazine
The occasion was a conference on the work of Shinichi Mochizuki, a brilliant mathematician at Kyoto University who in August 2012 released four papers that were both difficult to understand and impossible to ignore.
He called the work "inter-universal Teichmüller theory" (IUT theory) and explained that the papers contained a proof of the abc conjecture, one of the most spectacular unsolved problems in number theory.
Mochizuki had developed IUT theory over a period of nearly 20 years, working in isolation.
As a mathematician with a track record of solving hard problems and a reputation for careful attention to detail, he had to be taken seriously.
Yet his papers were nearly impossible to read.
The papers, which ran to more than 500 pages, were written in a novel formalism and contained many new terms and definitions.
Compounding the difficulty, Mochizuki turned down all invitations to lecture on his work outside of Japan.
"People are getting impatient, including me, including [Mochizuki], and it feels like certain people in the mathematical community have a responsibility to do something about this," Kim said.
"We do owe it to ourselves and, personally as a friend, I feel like I owe it to Mochizuki as well."
Shinichi Mochizuki appearing via videoconference to answer questions.
Shinichi Mochizuki appearing via videoconference to answer questions.
Philipp Ammon for Quanta Magazine
Until Mochizuki released his work, little progress had been made towards proving the abc conjecture since it was proposed in 1985.
Mochizuki employed a similar strategy in his work on abc.
Rather than proving abc directly, he set out to prove Szpiro's conjecture.
And to do so, he first encoded all the relevant information from Szpiro's conjecture in terms of a new class of mathematical objects of his own invention called Frobenioids.
Before Mochizuki began working on IUT theory, he spent a long time developing a different type of mathematics in pursuit of an abc proof.
He called that line of thought "Hodge-Arakelov theory of elliptic curves.
Go Yamashita lecturing on the work of Shinichi Mochizuki.
Go Yamashita lecturing on the work of Shinichi Mochizuki.
Philipp Ammon for Quanta Magazine
Mochizuki expressed much of the data from Szpiro's conjecture-which concerns elliptic curves-in terms of Frobenioids.
Just as Wiles moved from Fermat's Last Theorem to elliptic curves to Galois representations, Mochizuki worked his way from the abc conjecture to Szpiro's conjecture to a problem involving Frobenioids, at which point he aimed to use the richer structure of Frobenioids to obtain a proof.
"From Mochizuki's point of view, it's all about looking for a more fundamental reality that lies behind the numbers," Kim said.
In presentations at the end of the third day and first thing on the fourth day, Kiran Kedlaya, a number theorist at the University of California, San Diego, explained how Mochizuki intended to use Frobenioids in a proof of abc.
The understanding that Mochizuki had recast abc in terms of Frobenioids was a surprising and intriguing development.
These techniques appear in Mochizuki's four IUT theory papers, which were the subject of the last two days of the conference.
The job of explaining those papers fell to Chung Pang Mok of Purdue University and Yuichiro Hoshi and Go Yamashita, both colleagues of Mochizuki's at the Research Institute for Mathematical Sciences at Kyoto University.
"The reason it fell apart is not meant as a reflection of anything with Mochizuki," he said.
Others think the onus remains on Mochizuki to better explain his work.
"[I] got the impression that unless Mochizuki himself writes a readable paper, the matter will not be resolved," Faltings said by email.

next.dbrl.org [cached]

There's been a great deal of excitement in the math world recently, as Shinichi Mochizuki, a mathematician at Kyoto University in Japan, has recently released a 500-page proof of the abc conjecture, which proposes a relationship between prime numbers (numbers ...

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www.sciencenews.org

Six months after Shinichi Mochizuki of Kyoto University in Japan released his 500-page proof of the abc conjecture, that vetting process has yet to occur.

But this time, no one except Mochizuki seems to have any glimmering of how his proof works.
It is so peculiar that mathematicians might have dismissed it as the work of a crank, except that Mochizuki is known as a deep thinker with a record of strong results.
Also, they really hope he is right.
In August, when Mochizuki released the four papers explaining his proof, a number of mathematicians dove into them eagerly.
But they couldn't even understand his vocabulary.
Mochizuki had built an entirely new mathematical field, one he named "inter-universal Teichmüller geometry," and populated it with objects no one had ever heard of: "anabelioids," "Frobenoids," "NF-Hodge theaters.
Mathematicians began clamoring for Mochizuki to explain the kernel of his ideas, but he refused.
I suspect that this is the psychology of the situation for Mochizuki.
He said what he wanted to say in the paper."
Mochizuki has, however, been happy to answer specific questions by e-mail.
And recently he released a "panoramic overview," though many mathematicians find it nearly as impenetrable as the full proof.
Rumors circulated that some of the leading figures in the field had become skeptical of the proof.
Out of respect for Mochizuki - and hope that his proof will, in the end, turn out to be right - an effort was made to keep these rumors from circulating on the Internet, and no mathematician would go on the record expressing them.
Still, the rumors have further dampened enthusiasm for the hard work of slogging through the four papers.
Hope rests on a couple of Japanese mathematicians believed to be talking through the proof with Mochizuki, but they are unwilling to talk to the press.
"What we need is for this stuff to exist in someone else's brain besides [Mochizuki's].

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