The mystery concerns an impenetrable but potentially groundbreaking proof — of a puzzle known as the abc conjecture — that appeared online three years ago. Whether the proof is valid is still not clear — a source of frustration for some of the leading specialists who gathered at the University of Oxford on 7–11 December to discuss the matter.
Others say that the workshop, in which the proof’s reclusive architect Shinichi Mochizuki made a rare, virtual appearance, has at least boosted prospects for a resolution.
The quest to understand Mochizuki’s proof dates back to August 2012, when he quietly posted four papers on his website in which he claimed to have solved the abc conjecture. The problem gets its name from expressions of the form a + b = c and connects the prime numbers that are factors of a and b with those that are factors of c. Its solution could potentially change the face of number theory, which deals with the fundamental properties of, and relationships between, whole numbers.
Here is the first part of his 500 page magnum opus in case anyone wants to take a crack at verification: http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20I.pdf
Trivial stuff, really.
I’d do it myself, but the guy who developed much of the underlying theory was a hardcore Nazi, and I for one am not going to continue the work of a Nazi. Sticking by ma principles, I am.
Anyhow, flippancy aside, this is a very good demonstration of Apollo’s Ascent theory in action. The discovery threshold for in pure mathematics today is very high. At least 160, probably 175.
Mochizuki is clearly very, very bright.
- Attended Phillips Exeter Academy, regularly rated in the top 10 schools in the US (sometimes first).
- Entered Princeton University at the age of 16.
- Graduated 3 years later as a salutatorian.
- Got his PhD at the age of 23 under the supervision of a Fields Medal winner.
- Become professor a decade later.
So there are very few people in a position to even understand the stuff at the furthest frontiers.
“The decision about which topics to cover lacked some overall understanding of the proof,” says Jakob Stix, a number theorist at the Goethe University in Frankfurt, Germany. “Which is not really a complaint, because I sense that nobody really understands the proof.” …
Some, such as Felipe Voloch of the University of Texas at Austin, were more scathing. “The play showing today at the Hodge Theatre was a farce,” Voloch wrote online, referring to a theoretical construction that Mochizuki named a Hodge Theatre.
Attendees also restated familiar complaints about the proof itself. “The amount of language seems absurd,” said Artur Jackson of Purdue University in West Lafayette, Indiana, at the end of Thursday. And, Voloch told Nature: “I don’t know why he chose to make it so abstract.”
A follow-up workshop is expected to take place in Kyoto in July. Kedlaya plans to attend, unlike some of the disillusioned participants in the Oxford workshop. “The claim is an extremely important result,” he says, and the community deserves to know whether it is valid — even though the process will take several more years.
A couple further points.
Mochizuki is possibly the brightest mathmo on Earth, and he is Japanese (two other really big contemporary giants are the Chinese-American Terence Tao, and the Russian Jewish Perelman). Japan is the only major East Asian nation to have been developed for more than a generation. Clearly, pure IQ triumphs over any “q factor” let alone hazy Jaychickean notions of “clannishness” so far as intellectual accomplishment is concerned.
Second… We can envision some of the technologies that might conceivably play a big role in the coming century: Intelligence genes, geoengineering, SENS, autonomous robotics, self-assembling nanotech, superintelligence… It might be germane for serious futurists to start serious study of the sort of intelligence – loosely, in terms of standard deviations – that would be needed to actually discover them.