By Davide Castelvecchi and Nature magazine | October 8, 2015 — A Japanese mathematician claims to have solved one of the most important problems in his field. The trouble is, hardly anyone can work out whether he’s right.
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Sometime on the morning of August 30 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 bombshell. 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 century 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 University 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 invented a new branch of his discipline, one that is astonishingly abstract even by the standards of pure maths. “Looking at it, you feel a bit like you 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 graduate 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 Belgium, 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 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 mathematicians as a whole often face in communicating their craft to the wider world.
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 cannot be further factored out into smaller whole numbers: for example, 15 = 3 × 5 or 84 = 2 × 2 × 3 × 7. In principle, the prime factors of a and bhave 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 conjecture, 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 University in Cambridge, Massachusetts, realized that the abc conjecture, if true, would have profound implications for the study of equations concerning 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: confirmation of a conjecture formulated in 1922 by the US mathematician 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 teaching 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 department 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 mathematician 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 student couldn’t have been easy. “Faltings was at the top of the intimidation ladder,” 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 mathematician 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”.