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S
ometime on the morning of 30 August 2012, Shinichi Mochizuki
quietly posted four papers on his website.
The papers were huge — more than 500 pages in all — packed
densely with symbols, and the culmination of more than a decade of
solitary work. They also had the potential to be an academic bomb-
shell. In them, Mochizuki claimed to have solved the abc conjecture,
a 27-year-old problem in number theory that no other mathematician
had even come close to solving. If his proof was correct, it would be
one of the most astounding achievements of mathematics this cen-
tury and would completely revolutionize the study of equations with
whole numbers.
Mochizuki, however, did not make a fuss about his proof. The
respected mathematician, who works at Kyoto University’s Research
Institute for Mathematical Sciences (RIMS) in Japan, did not even
announce his work to peers around the world. He simply posted the
papers, and waited for the world to find out.
Probably the first person to notice the papers was Akio Tamagawa,
a colleague of Mochizuki’s at RIMS. He, like other researchers, knew
that Mochizuki had been working on the conjecture for years and had
been finalizing his work. That same day, Tamagawa e-mailed the news
to one of his collaborators, number theorist Ivan Fesenko of the Univer-
sity of Nottingham, UK. Fesenko immediately downloaded the papers
and started to read. But he soon became “bewildered”, he says. “It was
impossible to understand them.”
Fesenko e-mailed some top experts in Mochizuki’s field of arithmetic
geometry, and word of the proof quickly spread. Within days, intense
chatter began on mathematical blogs and online forums (see Nature
http://doi.org/725; 2012). But for many researchers, early elation about
the proof quickly turned to scepticism. Everyone — even those whose
area of expertise was closest to Mochizuki’s — was just as flummoxed by
the papers as Fesenko had been. To complete the proof, Mochizuki had
inventedanewbranchofhisdiscipline,onethatisastonishinglyabstract
evenbythestandardsofpuremaths.“Lookingatit,youfeelabitlikeyou
might be reading a paper from the future, or from outer space,” number
theorist Jordan Ellenberg, of the University of Wisconsin–Madison,
wrote on his blog a few days after the paper appeared.
Three years on, Mochizuki’s proof remains in mathematical
limbo — neither debunked nor accepted by the wider community.
Mochizuki has estimated that it would take an expert in arithmetic
geometry some 500 hours to understand his work, and a maths gradu-
ate student about ten years. So far, only four mathematicians say that
they have been able to read the entire proof.
Adding to the enigma is Mochizuki himself. He has so far lectured
about his work only in Japan, in Japanese, and despite being fluent
in English, he has declined invitations to talk about it elsewhere. He
does not speak to journalists; several requests for an interview for this
story went unanswered. Mochizuki has replied to e-mails from other
mathematicians and been forthcoming to colleagues who have visited
him, but his only public input has been sporadic posts on his website.
In December 2014, he wrote that to understand his work, there was a
“need for researchers to deactivate the thought patterns that they have
installed in their brains and taken for granted for so many years”. To
mathematician Lieven Le Bruyn of the University of Antwerp in Bel-
gium, Mochizuki’s attitude sounds defiant. “Is it just me,” he wrote on
his blog earlier this year, “or is Mochizuki really sticking up his middle
finger to the mathematical community”.
Now, that community is attempting to sort the situation out. In
December, the first workshop on the proof outside of Asia will take
Shinichi Mochizuki claims
to have solved one of the
most important problems in
mathematics. The trouble is,
hardly anyone can work out
whether he’s right.
The
impenetrable
proof
By Davide Castelvecchi
ILLUSTRATIONBYPADDYMILLS FEATURE NEWS
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place in Oxford, UK. Mochizuki will not be there in person, but he
is said to be willing to answer questions from the workshop through
Skype. The organizers hope that the discussion will motivate more
mathematicians to invest the time to familiarize themselves with his
ideas — and potentially move the needle in Mochizuki’s favour.
In his latest verification report, Mochizuki wrote that the status of
his theory with respect to arithmetic geometry “constitutes a sort of
faithful miniature model of the status of pure mathematics in human
society”. The trouble that he faces in communicating his abstract work
to his own discipline mirrors the challenge that mathema-
ticians as a whole often face in communicating their
craft to the wider world.
Primal importance
The abc conjecture refers to numerical
expressions of the type a + b = c. The
statement, which comes in several
slightly different versions, concerns
the prime numbers that divide each of
the quantities a, b and c. Every whole
number, or integer, can be expressed in
an essentially unique way as a product
of prime numbers — those that can-
not be further factored out into smaller
whole numbers: for example, 15 = 3 × 5 or
84 = 2 × 2 × 3 × 7. In principle, the prime fac-
tors of a and b have no connection to those of
their sum, c. But the abc conjecture links them
together. It presumes, roughly, that if a lot of small
primes divide a and b then only a few, large ones divide c.
This possibility was first mentioned in 1985, in a rather off-hand
remark about a particular class of equations by French mathematician
Joseph Oesterlé during a talk in Germany. Sitting in the audience was
David Masser, a fellow number theorist now at the University of Basel
in Switzerland, who recognized the potential importance of the conjec-
ture, and later publicized it in a more general form. It is now credited to
both, and is often known as the Oesterlé–Masser conjecture.
A few years later, Noam Elkies, a mathematician at Harvard Univer-
sity in Cambridge, Massachusetts, realized that the abc conjecture, if
true, would have profound implications for the study of equations con-
cerning whole numbers — also known as Diophantine equations after
Diophantus, the ancient-Greek mathematician who first studied them.
Elkies found that a proof of the abc conjecture would solve a huge
collection of famous and unsolved Diophantine equations in one
stroke. That is because it would put explicit bounds on the size of the
solutions. For example, abc might show that all the solutions to an
equation must be smaller than 100. To find those solutions, all one
would have to do would be to plug in every number from 0 to 99 and
calculate which ones work. Without abc, by contrast, there would be
infinitely many numbers to plug in.
Elkies’s work meant that the abc conjecture could supersede the most
important breakthrough in the history of Diophantine equations: con-
firmation of a conjecture formulated in 1922 by the US mathemati-
cian Louis Mordell, which said that the vast majority of Diophantine
equations either have no solutions or have a finite number of them.
That conjecture was proved in 1983 by German mathematician
Gerd Faltings, who was then 28 and within three years would win a
Fields Medal, the most coveted mathematics award, for the work. But if
abc is true, you don’t just know how many solutions there are, Faltings
says, “you can list them all”.
Soon after Faltings solved the Mordell conjecture, he started teach-
ing at Princeton University in New Jersey — and before long, his path
crossed with that of Mochizuki.
Born in 1969 in Tokyo, Mochizuki spent his formative years in the
United States, where his family moved when he was a child. He attended
an exclusive high school in New Hampshire, and his precocious talent
earned him an undergraduate spot in Princeton’s mathematics depart-
ment when he was barely 16. He quickly became legend for his original
thinking, and moved directly into a PhD.
People who know Mochizuki describe him as a creature of habit
with an almost supernatural ability to concentrate. “Ever since he was
a student, he just gets up and works,” says Minhyong Kim, a mathema-
tician at the University of Oxford, UK, who has known Mochizuki
since his Princeton days. After attending a seminar or colloquium,
researchers and students would often go out together for
a beer — but not Mochizuki, Kim recalls. “He’s not
introverted by nature, but he’s so much focused
on his mathematics.”
Faltings was Mochizuki’s adviser for his
senior thesis and for his doctoral one, and
he could see that Mochizuki stood out. “It
was clear that he was one of the brighter
ones,” he says. But being a Faltings stu-
dent couldn’t have been easy. “Faltings
was at the top of the intimidation lad-
der,” recalls Kim. He would pounce on
mistakes, and when talking to him, even
eminent mathematicians could often be
heard nervously clearing their throats.
Faltings’s research had an outsized
influence on many young number theorists
at universities along the US eastern seaboard.
His area of expertise was algebraic geometry,
which since the 1950s had been transformed into
a highly abstract and theoretical field by Alexander
Grothendieck — often described as the greatest mathema-
tician of the twentieth century. “Compared to Grothendieck,” says
Kim, “Faltings didn’t have as much patience for philosophizing.” His
style of maths required “a lot of abstract background knowledge — but
also tended to have as a goal very concrete problems. Mochizuki’s work
on abc does exactly this”.
Single-track mind
After his PhD, Mochizuki spent two years at Harvard and then in
1994 moved back to his native Japan, aged 25, to a position at RIMS.
Although he had lived for years in the United States, “he was in some
ways uncomfortable with American culture”, Kim says. And, he adds,
growing up in a different country may have compounded the feeling
of isolation that comes from being a mathematically gifted child. “I
think he did suffer a little bit.”
Mochizuki flourished at RIMS, which does not require its faculty
members to teach undergraduate classes. “He was able to work on his
own for 20 years without too much external disturbance,” Fesenko says.
In 1996, he boosted his international reputation when he solved a con-
jecture that had been stated by Grothendieck; and in 1998, he gave an
invited talk at the International Congress of Mathematicians in Berlin
— the equivalent, in this community, of an induction to a hall of fame.
But even as Mochizuki earned respect, he was moving away from the
mainstream. His work was reaching higher levels of abstraction and he
was writing papers that were increasingly impenetrable to his peers. In
the early 2000s he stopped venturing to international meetings, and
colleagues say that he rarely leaves the Kyoto prefecture any more. “It
requires a special kind of devotion to be able to focus over a period of
many years without having collaborators,” says number theorist Brian
Conrad of Stanford University in California.
Mochizuki did keep in touch with fellow num-
ber theorists, who knew that he was ultimately
aiming for abc. He had next to no competition:
most other mathematicians had steered clear
of the problem, deeming it intractable. By early
“Looking at it,
you feel a bit
like you might
be reading a
paper from the
future.”
NATURE.COM
Tohearapodcaston
ShinichiMochizuki’s
proof,visit:
go.nature.com/v6rfy7
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2012, rumours were flying that Mochizuki was getting close to a proof.
Then came the August news: he had posted his papers online.
The next month, Fesenko became the first person from outside Japan
to talk to Mochizuki about the work he had quietly unveiled. Fesenko
was already due to visit Tamagawa, so he went to see Mochizuki too.
The two met on a Saturday in Mochizuki’s office, a spacious room
offering a view of nearby Mount Daimonji and with neatly arranged
books and papers. It is “the tidiest office of any mathematician I’ve
ever seen in my life”, Fesenko says. As the two mathematicians sat in
leather armchairs, Fesenko peppered Mochizuki with questions about
his work and what might happen next.
Fesenko says that he warned Mochizuki against speaking to the press
about his proof. He was mindful of the experience of another mathema-
tician: the Russian topologist Grigori Perelman, who shot to fame in
2003 after solving the century-old Poincaré conjecture (see Nature 427,
388; 2004) and then retreated and became increasingly estranged from
friends, colleagues and the outside world. Fesenko knew Perelman, and
thinks that his behaviour was a result of excessive media attention. But
Fesenko soon saw that the two mathematicians’ personalities could
not have been more different. Whereas Perelman was known for his
awkward social skills (and for letting his fingernails grow unchecked),
Mochizuki is universally described as articulate and friendly — if
intensely private about his life outside of work.
Normally after a major proof is announced, mathematicians read the
work — which is typically a few pages long — and can understand the
general strategy. Occasionally, proofs are longer and more complex,
and years may then pass for leading specialists to fully vet it and reach
a consensus that it is correct. Perelman’s work on the Poincaré conjec-
ture became accepted in this way. Even in the case of Groth-
endieck’shighlyabstractwork,expertswereabletorelate
most of his new ideas to mathematical objects they
were familiar with. Only once the dust has settled
does a journal typically publish the proof.
But almost everyone who tackled Mochi-
zuki’s proof found themselves floored. Some
were bemused by the sweeping — almost
messianic — language with which Mochi-
zuki described some of his new theoretical
instructions: he even called the field that
he had created ‘inter-universal geometry’.
“Generally, mathematicians are very hum-
ble, not claiming that what they are doing
is a revolution of the whole Universe,” says
Oesterlé, at the Pierre and Marie Curie Uni-
versity in Paris, who made little headway in
checking the proof.
The reason is that Mochizuki’s work is so far
removed from anything that had gone before. He is
attempting to reform mathematics from the ground up, starting
from its foundations in the theory of sets (familiar to many as Venn
diagrams). And most mathematicians have been reluctant to invest
the time necessary to understand the work because they see no clear
reward: it is not obvious how the theoretical machinery that Mochizuki
has invented could be used to do calculations. “I tried to read some of
them and then, at some stage, I gave up. I don’t understand what he’s
doing,” says Faltings.
Fesenko has studied Mochizuki’s work in detail over the past year,
visited him at RIMS again in the autumn of 2014 and says that he
has now verified the proof. (The other three mathematicians who say
they have corroborated it have also spent considerable time work-
ing alongside Mochizuki in Japan.) The overarching theme of inter-
universal geometry, as Fesenko describes it, is that one must look at
whole numbers in a different light — leaving addition aside and see-
ing the multiplication structure as something malleable and deform-
able. Standard multiplication would then be just one particular case
of a family of structures, just as a circle is a special case of an ellipse.
Fesenko says that Mochizuki compares himself to the mathematical
giant Grothendieck — and it is no immodest claim. “We had math-
ematics before Mochizuki’s work — and now we have mathematics
after Mochizuki’s work,” Fesenko says.
But so far, the few who have understood the work have struggled
to explain it to anyone else. “Everybody who I’m aware of who’s come
close to this stuff is quite reasonable, but afterwards they become
incapable of communicating it,” says one mathematician who did not
want his name to be mentioned. The situation, he says, reminds him
of the Monty Python skit about a writer who jots down the world’s
funniest joke. Anyone who reads it dies from laughing and can never
relate it to anyone else.
And that, says Faltings, is a problem. “It’s not enough if you have
a good idea: you also have to be able to explain it to others.” Faltings
says that if Mochizuki wants his work to be accepted, then he should
reach out more. “People have the right to be eccentric as much as they
want to,” he says. “If he doesn’t want to travel, he has no obligation. If
he wants recognition, he has to compromise.”
Edge of reason
For Mochizuki, things could begin to turn around later this year, when
the Clay Mathematics Institute will host the long-awaited workshop
in Oxford. Leading figures in the field are expected to attend, includ-
ing Faltings. Kim, who along with Fesenko is one of the organizers,
says that a few days of lectures will not be enough to expose the entire
theory. But, he says, “hopefully at the end of the workshop enough
people will be convinced to put more of their effort into reading the
proof”.
Most mathematicians expect that it will take many
more years to find some resolution. (Mochizuki has
said that he has submitted his papers to a journal,
where they are presumably still under review.)
Eventually, researchers hope, someone will
be willing not only to understand the work,
but also to make it understandable to oth-
ers — the problem is, few want to be that
person.
Looking ahead, researchers think that it
is unlikely that future open problems will
be as complex and intractable. Ellenberg
points out that theorems are generally sim-
ple to state in new mathematical fields, and
the proofs are quite short.
The question now is whether Mochizuki’s
proof will edge towards acceptance, as Perelman’s
did, or find a different fate. Some researchers see a
cautionary tale in that of Louis de Branges, a well-estab-
lished mathematician at Purdue University in West Lafayette,
Indiana. In 2004, de Branges released a purported solution to the
Riemann hypothesis, which many consider the most important open
problem in maths. But mathematicians have remained sceptical of that
claim; many say that they are turned off by his unconventional theo-
ries and his idiosyncratic style of writing, and the proof has slipped
out of sight.
For Mochizuki’s work, “it’s not all or nothing”, Ellenberg says. Even
if the proof of the abc conjecture does not work out, his methods and
ideas could still slowly percolate through the mathematical commu-
nity, and researchers might find them useful for other purposes. “I do
think, based on my knowledge of Mochizuki, that the likelihood that
there’s interesting or important math in those documents is pretty
high,” Ellenberg says.
But there is still a risk that it could go the other way, he adds. “I think
it would be pretty bad if we just forgot about it. It would be sad.” ■
Davide Castelvecchi is a reporter for Nature in London.
“I tried to read
some of them and
then, at some
stage, I gave up.”
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