Inter-universal
Teichmüller Theory:
Inside the Controversy
Worldsheet Action, No. 1
September 2025

James Douglas Boyd
Editor-in-Chief
One struggles to think of a controversy in contemporary mathematics more infamous than the dispute between Professor Mochizuki Shinichi and Professors Peter Scholze and Jacob Stix over the proof of the abc conjecture given in Mochizuki's four-part paper series on inter-universal Teichmüller theory (IUT). This controversy ensued after Scholze and Stix made a week-long trip to discuss the theory with Mochizuki at RIMS in Kyoto in 2018. Scholze and Stix presented an argument against the IUT approach in a manuscript entitled "Why abc is still a conjecture".
After the papers were published in 2021, I spent a long time reading them – at restaurants, waiting at the barber shop, on trains, etc. I eventually met Mochizuki in the spring of 2024 and was invited to visit RIMS for six weeks in the autumn of 2024, during which I interviewed Mochizuki for over 12 hours about various technical details in the IUT papers. This was the first time Mochizuki ever agreed to any kind of interview.
Popular opinion holds that IUT doesn't prove abc, due to Scholze and Stix's argument that one can derive a contradiction from the setup that IUT uses in its inter-universal approach to Diophantine inequalities. I expect that the proof strategy taken by the IUT papers will not be accepted. I unpack some of the history of the development of IUT and try to explain why I doubt that IUT will be remembered as proving abc.
I also explain why Mochizuki has opposed the Scholze-Stix argument: part of the reason is that their argument pertains to the IUT setup and thus doesn't get into the IUT algorithms, which are considered crucial to the papers by Mochizuki and of the greatest interest to him due to their relationship with anabelian geometry.
The interesting twist to the story is that some mathematicians in anabelian geometry and étale homotopy who are relatively disinterested in abc are taking an interest in IUT. Essentially, Mochizuki has a vision for anabelian geometry involving studying arithmetic fundamental groups under non-scheme-theoretic mappings.
Why these two parallel developments? The setup of IUT involves a log-theta-lattice of Hodge theaters, which are models of ring/scheme theory. The theta-link used in the IUT setup, and at the heart of the Scholze-Stix argument, induces set-theoretic contradictions without the assistance of different universes and labels. On the other hand, it's a non-scheme-theoretic mapping. So, Mochizuki develops algorithms that involve techniques like anabelian reconstruction and transporting abstract groups between Hodge theaters in the lattice. The Scholze-Stix argument doesn't get into the algorithms, and those interested in the abc proof are concerned about the use of labels and universes to suspend set-theoretic contradictions, as well as the contradiction that arises from the theta-link with labels removed. On the other hand, for an anabelian geometer, just looking at abstract group transport and additive structure reconstruction under non-scheme-theoretic mappings is interesting.
Mochizuki's research interest might be characterized as an "arithmetic Teichmüller theory" (or a Teichmüller theory for ring/scheme theory, rather than ℂ), where non-scheme-theoretic mappings are treated as Teichmüller dilations. This vision is not fully spelled out in the IUT papers; they never quite explain exactly what arithmetic Teichmüller theory is.
So, I make the argument that, although the abc proof is very likely to meet sustained rejection by the math community, there is some interest in the anabelian and étale-homotopic aspects of some of the math in IUT, and that making explicit, in general terms, what arithmetic Teichmüller theory is – independently of the IUT setup and also distinguished from, say, absolute anabelian geometry – is probably the best achievable scenario.
Executive Summary