Inter-Universal Samurai
(宇宙際侍)
I.
It has been leaked [1, 2] that Scholze (in collaboration with Stix) is working on verifying Mochizuki’s inter-universal Teichmüller theory. In the course of this work, and some time after personally visiting Mochizuki in Japan, Scholze finds some critical missing element in Mochizuki’s proof and kindly sends the preprint to Mochizuki before releasing it. Mochizuki kindly welcomes the draft, asks for some time, and eventually produces a further long update of his work, sending it back to Scholze, together with a kind email. Scholze kindly asks for some time, and eventually kindly replies to Mochizuki that in the newest version of the argument there is another mistake, points it out, attaching an updated version of his manuscript. Mochizuki kindly welcomes it, asks for some more time, and eventually sends one more manuscript to Scholze. Kind exchange of emails follows, and after some work Scholze again sends an email, stating that he found a mistake, kindly points it out to Mochizuki, with a new version of a draft attached. Situation repeats again, and at the end of the next loop Mochizuki kindly asks Scholze if, instead entering further iteration of the loop, he would be kind to welcome a student of Mochizuki, Mr. Shisūji Nokami, who is well educated in the inter-universal theory, including most recent Mochizuki’s error corrections, and the reasoning that is behind it. Traditionally to Mochizuki [3], his email contains an exact specification of the amount of hours Mr. Shisūji spent completely dedicated to studying the theory, as well as an exact counting of how many times he has read through the particular manuscripts, including Scholze’s work. Scholze kindly agrees to welcome and talk with the visiting student. After some time Mr. Shisūji arrives to Bonn. On the next day after arrival, Scholze meets with Shisūji at Department of Mathematics. Shisūji is very kind and intelligent. He says that it is his first time in life he left Japan, and he shares his impressions how different are so many things in Bonn, as compared with Kyōto. After taking a coffee from department’s canteen, they go to some empty classroom, to discuss the current state of art in the inter-universal Teichmüller theory. When the door closes, and they both come to the blackboard, Shisūji says “it was my uttermost honour to meet you, Scholze-san”, and kindly pulls the detonator.
[1] Fesenko I., 2018, Remarks on Aspects of Modern Pioneering Mathematical Research, https://www.maths.nottingham.ac.uk/personal/ibf/rapm.pdf.
[2] Woit P., 2018, abc News, https://www.math.columbia.edu/~woit/wordpress/?p=10436.
[3] Mochizuki S., 2014, On the verification of inter-universal Teichmüller theory: a progress report (as of December 2014), http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTeich%20Verification%20Report%202014-12.pdf.
Warszawa, 1.8.18 (10:41a.m. GMT+02:00)*
ps. According to Marcin Kotowski (private communication, August 2018): «the detonator was not Mochizuki's idea, it was Shisūji's. He too came to believe there is a fatal mistake in the proof and decided to save the honor of his master.»
II.
* Funny synchronicity: This piece of math science fiction was written and emailed to a few mathematician friends of mine several hours before an announcement, at International Congress of Mathematicians, that Peter Scholze (together with Caucher Birkar, Alessi Fegalli, and Akshay Venkatesh) is awarded with 2018 Fields Medal [4], without me even knowing that the Congress is happening at the same time. (From reading Peter Woit’s blog, I knew that Scholze is considered as a candidate for this year’s price, yet without the knowledge of timing of ICM.)
[4] IMPA (anonymously), 2018, Researchers from Germany, India, Iran and Italy take home the 2018 Fields Medal, https://impa.br/en_US/page-noticias/researchers-from-germany-india-iran-and-italy-take-home-the-2018-fields-medal/.
III.
ps.2.
[The following text is a follow-up to my comment on Peter Woit’s blog, posted after public release of the Scholze–Stix paper and Mochizuki’s commentaries. This follow-up, unlike the original comment, was not accepted by Peter, due to the reasons unknown to me (for example, it could be too long or too much offtopic).]
To avoid misinterpretation, I want to add that by bringing up the perspective of cultural anthropology, I wasn’t intending any deconstruction (there is already enough flavour of the nominalistic reductionist attitude in the Scholze–Stix paper). In the same sense, by being satirical in the above fictional story I was not intending to be sarcastic or ironic. Quite oppositely, it seems to me that Mochizuki dreams out a big dream, and while he may or may not be able to turn it into a technically sound exact result (by means of the default professional standards), the scale and dedication of his efforts are remarkable on their own.
In a wider perspective, maybe the dramatic story of inter-universal Teichmüller theory should be taken as one of the indications how the Grothendieck-style holistic approach to mathematics can be powerful but also elusive, prone to shifting from a fairly-tale to swamps, not only due to richness and complexity of the higher order categorical generalisations of various objects, but also due to easiness of speculative weakenings of diagrams ‘up to a fleeting sense of higher taste’, which may carry even the wealthiest minds into the no man’s land. For example, Serre’s commentary in 1986 letter to Grothendieck: «I particularly remember the rather disastrous state of SGA 5, where the authors got lost in masses of diagrams whose commutativity they were reduced to asserting without proof (up to sign, with a little optimism...); and these commutations were essential for the sequel» [5] sounds strikingly similar, both in the content and context, to the Scholze–Stix commentary «[Mochizuki] claimed that up to the “blurring” given by certain indeterminacies the diagram does commute» [6].
The cultural differences may prevent someone to effectively communicate a valid project in a way that would be inviting enough so the critical mass of others would join in. E.g., Grothendieck knew how to do it by formulating standard conjectures, which eventually have lead to Deligne’s proof of Weyl’s riemannian conjecture. While Deligne didn’t rely on standard conjectures, nevertheless Grothendieck, by communicating his project in a way meaningful for Deligne (who was his student and had taken the innovative approach), is recognised as a person who essentially contributed to solving this problem. What if Mochizuki is strongly communicating his belief in reasonability of the project he got involved in, but lacks some sort of ability to form sufficiently sound ‘inter-universal standard conjectures’, and so he recourses to another (anecdotal, yet mind-boggling) forms of persuading the coherence of meaning he sees? After all, why he would even bother himself to put the Scholze–Stix paper on his own website? I referred to ‘samurai ethics’ because of the very specific aesthetic style Mochizuki himself chooses to communicate and stress the meaningfulness of the project he offers.
In this context, I’d like to recall the case of Minoru Tomita. His research into the structure of the standard forms of von Neumann algebras was a key breakthrough in the field, comparable (in the degree of revolutionary impact it brought) with the founding research by von Neumann and Murray. In particular, it has allowed classification of type III von Neumann algebras [7], the development of full-fledged theory of noncommutative integration [8, 9], as well as several deep results in mathematical physics (e.g. the Bisognano–Wichmann theorem [10] on emergence of space-time structure from the operator algebra of quantum fields, and the exact derivation of Hawking’s temperature formula based on relationship between modular automorphisms and the Kubo–Martin–Schwinger states [11, 12]). However, the original results presented by Tomita were technically flawed and became heavily criticised. It is very unclear what was Tomita’s own attitude to these mistakes. Only due to the further work by Masamichi Takesaki (by that time already a well-recognised researcher in the field of operator algebras), resulting in his book “Tomita’s theory of modular Hilbert algebras and its applications” [13], published three years later, the theory was accepted, and is now known as the Tomita–Takesaki theory. (There exists a valuable Takesaki’s historical account [14] on early reception and correction of Tomita’s results, and I draw here also upon some knowledge obtained by personal communication from other sources.) Tomita’s original work [15] remained unpublished and has been completely left aside, beyond the scientific circulation. It seemed to me always very strange, almost paradoxal, that the major paper in the field, even if being involved in a large amount of technical mistakes, has received so little credit and virtually no review with a commentary (except Takesaki’s book), e.g., as something like a ‘golden oldie’ (which is a common practice in the field of general relativity [16]), or as an appendix in some conference proceedings (due to its definite historical value). Before his “standard forms” work, Tomita had obtained several other valuable results in the same field, which were published and well-received within the cumulative time span of over a decade. Thus, his preprint definitely was not a miraculously opportune mistake of some shady individual, but a result of a long-term innovative work of a dedicated researcher. The denial of attribution of a positive value to the original breakthrough manuscript, even if taking into account the harshness of professional standards of the hard-core operator algebraic community, is noticeable on its own. Quite characteristically, the misnaming of Tomita’s manuscript is perpetuated throughout all of subject’s literature [17], showing clearly that this text was not only rejected in its validity, but also not seen at all by the experts. Furthermore, as opposed to his earlier works, there was practically none of later work by Tomita that would gain any noticeable impact in the field of operator algebras. This suggests a possible interpretation that the “forget Tomita’s paper” effect can be a socially constructed ‘losing the face’ (more specifically, a reaction of a denial by the community focused on high standards of mathematical exactness, when faced with the revolutionary result provided in a form that defects these standards). The specificity of Tomita’s personality quite probably contributed to this as well (I’ve heard a first-hand story about a «strange behaviour» of him while giving a talk at some conference – in particular, stating a specific lemma, together with a remark that he proved it many years ago but... forgot it, – which, however, makes more sense in the context of Connes’ remark: «he’s someone who has succeeded in avoiding all the traps that society tends to set for someone extremely original. He became deaf at the age of two. When he started his research, his thesis advisor gave him a huge book telling him, “Come back and see me once you have read this book”. Tomita met accidentally his thesis advisor two years later and the latter asked him, “How is the book going?” to which Tomita replied, “Oh, I lost it after one week”» [18]). While Mochizuki’s case has clear differences with respect to this story, it also shares certain stylistic similarity, which may represent some important characteristics of the contextual limitations of expression and communication of mathematical ideas.
To sum up these reflections: sometimes there are nontrivial issues in the cross-cultural communication that are also influencing mathematics (Ramanujan’s story being the clear-cut example), and maybe Mochizuki needs his own Takesaki (thus, someone who will be willing to invest essentially more than 5 days of discussion plus preparation and write-up) to make the inter-universal dreams Wick re-rotated into intersubjective realms (either as a sound proof of whatever is behind the “Corollary 3.12” or, at least, to settle some nontrivial ‘inter-universal conjectures’ for others to ponder upon by their own ways and dreams, hopefully ending in a real-life stream of proofs and publications). Maybe this is something that Mochizuki is communicating to us through his style, if we will take into account his (clearly manifested) confinement to specific ethics and aesthetics?
For sure, it is a quite speculative reading. In particular, positioning the analysis of Mochizuki’s anomaly inside the interpolation scale between Grothendieck’s and Tomita’s cases may be as well just a subjective locus of view, expressing my own mathematical interests more then anything real about Mochizuki. Yet, it seems to be likely that skilled visionaries are not always skilled translators, and then it is a matter of the socially constructed context, whether and how the adequate arrangements for translation can be successfully established. It does not seem to have happened so far at the scene of the inter-universal theatre.
[5] Colmez P., Serre J.-P. (eds.), 2004, Grothendieck–Serre correspondence. Bilingual edition, American Mathematical Society, Providence.
[6] Scholze P., Stix J., 2018, Why abc is still a conjecture, http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pdf.
[7] Takesaki M., 1979, 2003, Theory of operator algebras, Vol.1–3, Springer, Berlin.
[8] Haagerup U., 1979, Lp-spaces associated with an arbitrary von Neumann algebra, in: Algèbres d’opérateurs et leurs applications en physique mathématique (Proc. Colloques Internationaux, Marseille 20-24 juin 1977), Colloques Internationaux C.N.R.S. 274, Éditions du C.N.R.S., Paris, pp.175–184, http://dmitripavlov.org/scans/haagerup.pdf.
[9] Falcone A.J., Takesaki M., 2001, The non-commutative flow of weights on a von Neumann algebra, J. Funct. Anal. 182, 170–206, http://www.math.ucla.edu/~mt/papers/QFlow-Final.tex.pdf.
[10] Bisognano J.J., Wichmann E.H., 1975, On the duality condition for a hermitian scalar field, J. Math. Phys. 16, 985–1007; 1976, On the duality condition for quantum fields, J. Math. Phys. 17, 303–321.
[11] Sewell G.L., 1982, Quantum fields on manifolds: PCT and gravitationally induced thermal states, Ann. Phys. 141, 201–224.
[12] Fredenhagen K., Haag R., 1990, On the derivation of Hawking radiation associated with the formation of a black hole, Commun. Math. Phys. 127, 273–284.
[13] Takesaki M., 1970, Tomita's theory of modular Hilbert algebras and its applications, Springer, Berlin.
[14] Takesaki M., 2014, Structure of von Neumann algebras of type III, https://www.imsc.res.in/~sunder/mtnotes.pdf.
(Particularly relevant is the following fragment: «So the theory for a von Neumann algebra of type III was badly needed when Tomita proposed his theory at the Baton Rouge Meeting in the spring of 1967. But his preprint was very poorly written and full of poor mistakes: nobady bothers to check the paper. When I wrote to Dixmier in the late spring of 1967 about the validity of Tomita's work, he responded by saying that he was unable to go beyond the third page and mentioned that it was very improtant to decide the validity of Tomita's work. At any rate, Tomita's work was largely ignored by the participants of the Baton Rouge Conference. For the promiss I made to Hugenholtz and Winnink, I started very seriously in April, 1967, after returning from the US and was able to resque all the major results: not lemmas and small propositions, many of which are either wrong or nonsense. Then I spent the academic year of 1968 through 1969 at Univ. of Pennsylvania, where R.V. Kadison, S. Sakai, J.M. Fell, E. Effros, R.T. Powers, E. Størmer and B. Vowdon were, but non of them believed Tomita's result. So I checked once more and wrote a very detailed notes which was later published as Springer Lecture Notes No.128: simplification was not an issue, but the validity of Tomita's claim. Through writing up the notes, I dicovered that Tomita's work could go much further than his claim: the modular condition, (called the KMS-condition by physicists), and a lot more. I know that the crossed product of a von Neumann algebra by the modular automorphism group is semi-finite which I didn't include in the lecture notes because I thought that the semi-finiteness alone was a half cocked claim. When I mentioned firmly the validity of Tomita's claim, the people at the U. of Pennsylvania decided to run an inspection seminar in which I was allowed to give only the first introductory talk, but not in
subsequent seminars, which run the winter of 1969 through the spring and the validity was established at the end.»)
[15] Tomita M., 1967, Standard forms of von Neumann algebras, Kyūshū University, Fukuoka, http://www.fuw.edu.pl/~kostecki/scans/tomita1967.pdf.
[16] MacCallum M. (ed.), General Relativity and Gravitation Golden Oldies, https://www.springer.com/gp/livingreviews/relativity/grg-golden-oldies.
[17] The preprint [15] is misreferenced in M. Takesaki’s books [13] and [7], as well as in all publications of other authors (with a single exception of [19]), as “Quasi-standard von Neumann algebras”, yet it is properly referenced in Tomita’s own paper, von Neumann 代数の標準型について [On standard forms of von Neumann algebras], 1967, published in: 第5回関数解析シンポジウム [Fifth Functional Analysis Symposium], Tōhoku University, Sendai, pp.101–102.
[18] Connes A., Goldstein C., Skandalis G., 2007, An interview with Alain Connes. Part I, Newsletter EMS 63, 25–31, http://alainconnes.org/docs/Inteng.pdf.
[19] Tomiyama J., 1969, On the tensor products of von Neumann algebras, Pacific J. Math. 30, 263–270.
(Ħaġar Qim/San Ġiljan, 秋分の日 23.09.2018)