"Vesselin Dimitrov, a graduate student at Yale University, has been concentrating on reading Mochizuki’s preliminary writing as preparation for reading the proof. In a series of e-mails, he explained that he’s drawn by both the challenge of the ABC conjecture and the elegance of Mochizuki’s thinking. “Reading through Mochizuki’s world,” Dimitrov writes, “I am much impressed by the unity and structural coherence that it exhibits. “
Dimitrov stressed that it’s too early to predict whether Mochizuki’s proof will stand up to the intense scrutiny coming its way. In October he and a collaborator, Akshay Venkatesh at Stanford University, sent a letter to Mochizuki about an error they found in the third and fourth papers of the proof. In response, Mochizuki posted a reply to his website acknowledging the error but explaining that it was minor and didn’t affect his conclusions. He is expected to post a corrected version of his proof by January."
The math community has reacted to Mochizuki’s proof with equal parts hope and skepticism, though few mathematicians are willing to discuss their doubts on the record out of respect for Mochizuki and a desire not to prejudge the vetting process. Mathematicians speak of a “brick wall” in mathematical reasoning that has thwarted previous attempts to solve ABC. “Before Mochizuki came along, this problem was viewed as utterly, hopelessly intractable and out of reach, like an out-of-the-solar-system kind of situation,” says Lagarias. In that light, any supposed proof was bound to be greeted with some doubt. Another source of skepticism is the potential expansiveness of Mochizuki’s accomplishment. It has long been understood by mathematicians that any proof of ABC would have the effect of simultaneously proving four other theorems (the work of Roth, Baker, Faltings, and Wiles) that stand among the most celebrated achievements in math in the last half-century. If Mochizuki has found a way to subsume those monumental results into a single formula, his work would take its place alongside equations like Einstein’s E=mc2 and the inequality behind Heisenberg’s uncertainty principle in terms of its sheer explanatory power. To many, such a discovery seems too good to be possible. Minhyong Kim thinks that the initial reaction to Mochizuki’s work owes to something else. “Frankly, there are many people who express skepticism because they look at it and they can’t understand what’s going on, and of course when you can’t understand something the most natural initial response is to be skeptical.”
In the coming weeks mathematicians are hoping Mochizuki will provide a kind of “executive summary” of his work―a 20- or 30-page template that traces the pivot points in his logic. Meanwhile, mathematicians like Ellenberg and Dimitrov will continue to poke at Mochizuki’s proof, looking for openings, raising questions, and translating Mochizuki’s ideas into terms that a wider circle of mathematicians can understand.
If Mochizuki’s work makes it through these informal early checks, work groups and conferences will be organized around his ideas. The Clay Mathematics Institute at Oxford has already expressed interest in sponsoring one such workshop. Further down the line―perhaps a year from now, if Mochizuki is able to build up sufficient trust with his peers―Mochizuki will submit his work for journal publication and it will be sent out for peer review. Mochizuki’s reputation as a gifted mathematician will survive even if his proof turns out to be wrong. But the same cannot be said for his work of the last decade. Mathematicians are tantalized by the possibility of a proof of ABC, but if an error is found early in the vetting process, the math world will likely move on without bothering to explore the rest of the mathematical universe Mochizuki has created. Mochizuki has almost certainly made significant discoveries―but without the allure of a proof, it’s possible no one will take the time to understand them.
Mochizuki, who is known as a shy person and has declined interview requests since publishing the proof in August, will have an important role to play in all of this, answering queries and explaining his work to a math community from which he long ago parted ways. And this, Kim thinks, might pose the greatest challenge of all. “When you’ve been wrapped up in your own research program for a long time sometimes you lose a sense of what it is that other people don’t understand,” he says. “Other people feel quite mystified as to what he’s doing and part of him, I suspect, doesn’t quite understand why.”
