abc

In order to understand why Mochizuki’s situation is unusual, it’s useful to compare it to the two biggest discoveries in math in the last 20 years: the British number theorist Sir Andrew Wiles’s proof of Fermat’s Last Theorem (he was knighted for his accomplishment) in 1995 and Russian mathematician Grigori Perelman’s proof of the Poincaré conjecture in 2003. Hailed as singular discoveries, these proofs transformed their authors into the two most famous mathematicians in the world (alongside, perhaps, John Nash of “A Beautiful Mind” fame). But their work built from a base of well-understood mathematics, following routes that others had already speculated could lead to proofs. As a result, the mathematical community was able to verify Wiles’s and Perelman’s proofs in relatively short order.

Not so with Mochizuki’s proof.

Mochizuki made a name for himself in his 20s based on a number of significant contributions to a relatively new, complex subfield of arithmetic geometry known as anabelian geometry. In 1998 he received one of the highest honors in math—an invitation to address the quadrennial International Congress of Mathematicians. He gave his talk in August 1998 in Berlin, then effectively went to ground, disappearing for 14 years to work on ABC.

“I guess he wanted to work on a problem worthy of putting his full powers on,” says Jeffrey Lagarias, a professor of math at the University of Michigan, “and the ABC conjecture fit beautifully.”

Before mathematicians can even start to read the proof, or understand his four papers, they need to wade through 750 pages of Mochizuki’s incredibly complicated foundational work in anabelian geometry. At the moment, there are only about 50 people in the world who know anabelian geometry well enough to understand this preliminary work. Then, the proof itself is written in an entirely different branch of mathematics called “inter-universal geometry” that Mochizuki—who refers to himself as an “inter-universal Geometer”—invented and of which, at least so far, he is the sole practitioner.