An Update on Shinichi Mochizuki, Inter-Universal Teichmüller Theory, and the ABC Conjecture

The occasion was a conference on the work of Shinichi Mochizuki, a brilliant mathematician at Kyoto University who in August 2012 released four papers that were both difficult to understand and impossible to ignore. He called the work “inter-universal Teichmüller theory” (IUT theory) and explained that the papers contained a proof of the abc conjecture, one of the most spectacular unsolved problems in number theory.

Within days it was clear that Mochizuki’s potential proof presented a virtually unprecedented challenge to the mathematical community. Mochizuki had developed IUT theory over a period of nearly 20 years, working in isolation. As a mathematician with a track record of solving hard problems and a reputation for careful attention to detail, he had to be taken seriously. Yet his papers were nearly impossible to read. The papers, which ran to more than 500 pages, were written in a novel formalism and contained many new terms and definitions. Compounding the difficulty, Mochizuki turned down all invitations to lecture on his work outside of Japan. Most mathematicians who attempted to read the papers got nowhere and soon abandoned the effort.

For three years, the theory languished. Finally, this year, during the week of December 7, some of the most prominent mathematicians in the world gathered at the Clay Mathematical Institute at Oxford in the most significant attempt thus far to make sense of what Mochizuki had done. Minhyong Kim, a mathematician at Oxford and one of the three organizers of the conference, explains that the attention was overdue.

“People are getting impatient, including me, including [Mochizuki], and it feels like certain people in the mathematical community have a responsibility to do something about this,” Kim said. “We do owe it to ourselves and, personally as a friend, I feel like I owe it to Mochizuki as well.”

The conference featured three days of preliminary lectures and two days of talks on IUT theory, including a culminating lecture on the fourth paper, where the proof of abc is said to arise. Few entered the week expecting to leave with a complete understanding of Mochizuki’s work or a clear verdict on the proof.  What they did hope to achieve was a sense of the strength of Mochizuki’s work. They wanted to be convinced that the proof contains powerful new ideas that would reward further exploration.

Frobenioids work in much the same way as the group described above. Instead of a square, they are an algebraic structure extracted from a special kind of elliptic curve. Just as in the example above, Frobenioids have symmetries beyond those arising from the original geometric object. Mochizuki expressed much of the data from Szpiro’s conjecture — which concerns elliptic curves — in terms of Frobenioids. Just as Wiles moved from Fermat’s Last Theorem to elliptic curves to Galois representations, Mochizuki worked his way from the abc conjecture to Szpiro’s conjecture to a problem involving Frobenioids, at which point he aimed to use the richer structure of Frobenioids to obtain a proof.

In presentations at the end of the third day and first thing on the fourth day, Kiran Kedlaya, a number theorist at the University of California, San Diego, explained how Mochizuki intended to use Frobenioids in a proof of abc. His talks clarified a central concept in Mochizuki’s method and generated the most significant progress at the conference thus far.

“Hope Rekindled for Perplexing Proof” can be read here.

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