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.