“Research is what I’m doing when I don’t know what I’m doing.” — Wernher von Braun
I find Shinichi Mochizuki’s proof of the abc conjecture fascinating. Not the content of the proof—which I do not understand in the least—but the circumstances of the proof. Most mathematics, no matter how arcane it appears to outsiders, is not that original. Experts in a specific area can judge just how much or how little a new paper adds to their body of knowledge, and usually it’s not much. But Mochizuki’s work is such a departure from the mainstream that experts have been trying to understand it for four years now.
Five days ago, Nature published an article headlined Monumental Proof to Torment Mathematicians for Years to Come.
… Kedlaya says that the more he delves into the proof, the longer he thinks it will take to reach a consensus on whether it is correct. He used to think that the issue would be resolved perhaps by 2017. “Now I’m thinking at least three years from now.”
Others are even less optimistic. “The constructions are generally clear, and many of the arguments could be followed to some extent, but the overarching strategy remains totally elusive for me,” says mathematician Vesselin Dimitrov of Yale University in New Haven, Connecticut. “Add to this the heavy, unprecedentedly indigestible notation: these papers are unlike anything that has ever appeared in the mathematical literature.”
But today, New Scientist has an article with the headline Mathematicians finally starting to understand epic ABC proof. According to this article,
At least 10 people now understand the theory in detail, says Fesenko, and the IUT papers have almost passed peer review so should be officially published in a journal in the next year or so.
It’s interesting that the proof is so poorly understood that experts disagree on just how well it is understood.