17 December 2015

A collective effort to solve a recently discovered problem in mathematics has yielded few clues and no definitive answers.

The problem is entitled the abc conjecture (also known as the Oesterlé–Masser conjecture). It was first proposed by Joseph Oesterlé (1988) and David Masser (1985), and can be stated as follows: For every ε > 0, there are only finitely many triples of coprime positive integers a + b = c such that c > d^1+ε, where d denotes the product of the distinct prime factors of the product abc.

Approximately three years ago, an alleged proof for the abc conjecture appeared online. Written by Shinichi Mochizuki, the proof is deep and seemingly impenetrable, according to a team of leading specialists who had gathered at the University of Oxford on 7–11 December earlier this year to discuss the matter.

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 correct solution of this problem has profound implications for arithmetic geometry. But Mochizuki’s papers, which totalled more than 500 pages, were exceedingly abstract and cryptic.

Mochizuki, who is now 46 and a highly respected member of the Research Institute for Mathematical Sciences (RIMS) at Kyoto University in Japan, has rejected all invitations to lecture about his papers outside of Japan.

The Oxford workshop covered both Mochizuki’s preliminary work and an outline of his four abc papers. The contents of the papers were presented by two researchers who claim that they have checked the proof in its entirety — number theorists Yuichiro Hoshi and Go Yamashita, both from RIMS. The workshop was hosted by the Clay Mathematics Institute, a non-profit organization housed in the University of Oxford’s main mathematics building.

The highlight of the workshop was a lecture on 9 December by Kiran Kedlaya, an arithmetic geometer from the University of California, San Diego. He used a result from a 2008 paper by Mochizuki to demonstrate a crucial linkage between the abc conjecture and topology. However, the rest of the conference failed to build on this successful recognition.

A follow-up workshop is expected to take place in Kyoto in July.