Inter-universal Teichmüller theory

Inter-universal Teichmüller theory (abbreviated as IUT or IUTT) is the name given by mathematician Shinichi Mochizuki to a theory he developed in the 2000s, following his earlier work in arithmetic geometry. According to Mochizuki, it is "an arithmetic version of Teichmüller theory for number fields equipped with an elliptic curve". The theory was made public in a series of four preprints posted in 2012 to his website. The most striking claimed application of the theory is to provide a proof for various outstanding conjectures in number theory, in particular the abc conjecture. Mochizuki and a few other mathematicians claim that the theory indeed yields such a proof but this has so far not been accepted by the mathematical community.

History

The theory was developed entirely by Mochizuki up to 2012, and the last parts were written up in a series of four preprints.[1] Mochizuki then made his work public in 2012 in a rather unusual manner, only making the papers available on his RIMS webpage and avoiding announcements or posting to a prepublication server. Soon after, the papers were picked up by Ivan Fesenko and the mathematical community at large was made aware of the claims to have proven the abc conjecture.

The reception of the claim was at first enthusiastic, though number theorists were baffled by the original language introduced and used by Mochizuki.[2][3] National workshops on IUT were held at RIMS in March 2015 and in Beijing in July 2015.[4] International workshops on IUT were held in Oxford in December 2015 and at RIMS in July 2016. The international workshops attracted more than 100 participants. Presentations from these workshops are available online.[5][6] However, these did not lead to broader understanding of Mochizuki's ideas and the status of his claimed proof was not changed by these events.[7]

In 2017, a number of mathematicians who had examined Mochizuki's argument in detail pointed to a specific point which they could not understand, near the end of the proof of Corollary 3.12, in paper three of four.[8][9]

In March 2018, Peter Scholze and Jakob Stix visited Kyoto University for five days of discussions with Mochizuki and Yuichiro Hoshi; while this did not resolve the differences, it brought into focus where the difficulties lay.[8][10] It also resulted in the publication of reports of the discussion by both sides:

  • In May 2018, Scholze and Stix wrote a 10-page report, updated in September 2018, detailing the (previously identified) gap in Corollary 3.12 in the proof, describing it as "so severe that in [their] opinion small modifications will not rescue the proof strategy", and that Mochizuki's preprint cannot claim a proof of abc.[11] They make a number of simplifications of IUTT, some drastic and not all of which Mochizuki considers valid, and insist on the distinction he does not make between 'abstract and concrete "pilot objects"'.
  • In September 2018, Mochizuki wrote a 41-page summary of his view of the discussions and his conclusions about which aspects of his theory he considers misunderstood.[12] In particular he names:
    • "re-initialization" of (mathematical) objects, making their previous "history" inaccessible;
    • "labels" for different "versions" of objects;
    • the emphasis on the types ("species") of objects.
  • In July and October 2018, Mochizuki wrote 8- and 5-page reactions to the May and September versions of the Scholze and Jakob Stix report, maintaining that the gap is the result of their simplifications, and that there is no gap in his theory.[13][14]

The 2017 comments and 2018 discussions were described in an article in Quanta Magazine in September 2018.[8]

Mathematical significance

Scope of the theory

Inter-universal Teichmüller theory is a continuation of Mochizuki's previous work in arithmetic geometry. This work, which has been peer-reviewed and well-received by the mathematical community, includes major contributions to anabelian geometry, and the development of p-adic Teichmüller theory, Hodge–Arakelov theory and Frobenioid categories. It was developed with explicit references to the aim of getting a deeper understanding of abc and related conjectures. In the geometric setting, analogues to certain ideas of IUT appear in the proof by Bogomolov of the geometric Szpiro inequality.[15]

The key prerequisite for IUT is Mochizuki's mono-anabelian geometry and its powerful reconstruction results, which allows to retrieve various scheme-theoretic objects associated to an hyperbolic curve over a number field from the knowledge of its fundamental group, or of certain Galois groups. IUT applies algorithmic results of mono-anabelian geometry to reconstruct relevant schemes after applying arithmetic deformations to them; a key role is played by three rigidities established in Mochizuki's etale theta theory. Roughly speaking, arithmetic deformations change the multiplication of a given ring, and the task is to measure how much the addition is changed.[16] Infrastructure for deformation procedures is decoded by certain links between so called Hodge theaters, such as a theta-link and a log-link.[17]

These Hodge theaters use two main symmetries of IUT: multiplicative arithmetic and additive geometric. On one hand, Hodge theaters generalize such classical objects in number theory as the adeles and ideles in relation to their global elements. On the other hand, they generalize certain structures appearing in the previous Hodge-Arakelov theory of Mochizuki. The links between theaters are not compatible with ring or scheme structures and are performed outside conventional arithmetic geometry. However, they are compatible with certain group structures, and absolute Galois groups as well as certain types of topological groups play a fundamental role in IUT. Considerations of multiradiality, a generalization of functoriality, imply that three mild indeterminacies have to be introduced.[17]

Consequences in number theory

The main claimed application of IUT is to various conjectures in number theory, among them abc, but also more geometric conjectures such as Szpiro's conjecture on elliptic curves and Vojta's conjecture for curves.

The first step is to translate arithmetic information on these objects to the setting of Frobenioid categories. It is claimed that extra structure on this side allows one to deduce statements which translate back into the claimed results.[18]

One issue with Mochizuki's arguments, which he acknowledges, is that it does not seem possible to get intermediate results in his proof of abc using IUT. In other words, there is no smaller subset of his arguments more easily amenable to an analysis by outside experts, which would yield a new result in Diophantine geometries.[18]

Vesselin Dimitrov extracted from Mochizuki's arguments a proof of a quantitative result on abc, which could in principle give a refutation of the proof.[19]

References
  1. Mochizuki, Shinichi (2012a), Inter-universal Teichmuller Theory I: Construction of Hodge Theaters (PDF)
    Mochizuki, Shinichi (2012b), Inter-universal Teichmuller Theory II: Hodge–Arakelov-theoretic Evaluation (PDF)
    Mochizuki, Shinichi (2012c), Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice (PDF)
    Mochizuki, Shinichi (2012d), Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations (PDF), archived from the original (PDF) on 2016-12-28, retrieved 2012-09-09
  2. Ball, Peter (10 September 2012). "Proof claimed for deep connection between primes". Nature. doi:10.1038/nature.2012.11378. Retrieved 19 March 2018.
  3. The Paradox of the Proof By Caroline Chen, accessed May 11, 2013
  4. Future and past workshops on IUT theory of Shinichi Mochizuki
  5. "Oxford Workshop on IUT Theory of Shinichi Mochizuki, December 7–11 2015". University of Nottingham. Retrieved 2018-03-19.
  6. "Inter-universal Teichmüller Theory Summit 2016 (RIMS workshop, July 18-27 2016)". University of Nottingham. Retrieved 2018-03-19.
  7. Revell, Timothy (December 18, 2017). "Mathematician set to publish ABC proof almost no one understands". New Scientist. Retrieved April 14, 2018.
  8. Klarreich, Erica (September 20, 2018). "Titans of Mathematics Clash Over Epic Proof of ABC Conjecture". Quanta Magazine.
  9. "The ABC conjecture has still not been proved". December 17, 2017. Retrieved March 17, 2018. For each of these people, the proof that had stumped them was for [Corollary] 3.12 in IUT3. It was striking to get three independent unsolicited emails in a matter of days which all zeroed in on that same proof as a point of confusion.
  10. Mochizuki, Shinichi. "March 2018 Discussions on IUTeich". Retrieved October 2, 2018. Web-page by Mochizuki describing discussions and linking consequent publications (following references), papers by Ivan Fesenko and a video by Fumiharu Kato of Tokyo Institute of Technology
  11. Scholze, Peter; Stix, Jakob. "Why abc is still a conjecture" (PDF). Retrieved September 23, 2018. (updated version of their May report)
  12. Mochizuki, Shinichi. "Report on Discussions, Held during the Period March 15 – 20, 2018, Concerning Inter-Universal Teichmüller Theory" (PDF). Retrieved October 2, 2018. the … discussions … constitute the first detailed, … substantive discussions concerning negative positions … IUTch.
  13. Mochizuki, Shinichi. "Comments on the manuscript by Scholze-Stix concerning Inter-Universal Teichmüller Theory" (PDF). Retrieved October 2, 2018.
  14. Mochizuki, Shinichi. "Comments on the manuscript (2018-08 version) by Scholze-Stix concerning Inter-Universal Teichmüller Theory" (PDF). Retrieved October 2, 2018. Most of the Comments of (his previous reaction) were not addressed in (their September update) and hence … remain valid Supplement to his previous reaction
  15. Mochizuki, Shinichi (2016), Bogomolov’s proof of the geometric version of the Szpiro conjecture from the point of view of inter-universal Teichmüller theory, Res. Math. Sci. 3(2016), 3:6
  16. Fesenko, Ivan (2016), Fukugen, Inference: International Review of Science, 2016
  17. Mochizuki, Shinichi (2016), The mathematics of mutually alien copies: from Gaussian integrals to inter-universal Teichmüller theory (PDF)
  18. Conrad, Brian (December 15, 2015). "Notes on the Oxford IUT workshop by Brian Conrad". 3. What is Inter-universal Teichmuller Theory (IUT)?. Retrieved March 18, 2018.CS1 maint: location (link)
  19. Vesselin, Dimitrov (14 January 2016). "Effectivity in Mochizuki's work on the abc-conjecture". arXiv:1601.03572.
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