Brief Report on the Current Situation
Surrounding Inter-universal Teichmüller Theory (IUT)
Shinichi Mochizuki
August 2023

Despite- the publication in 2021 of the four main papers [IUTchI], [IUTchII], [IUTchIII],[IUTchIV] on inter-universal Teichmüller theory (IUT) in a special volume of Publ. RIMS
edited by a special editorial board consisting of such internationally renowned
mathematicians as M. Kashiwara, A. Tamagawa, S. Mukai, H. Nakajima, and T.
Mochizuki;- the subsequent publication of the sequel [ExpEst] on explicit estimates in IUT in the Kodai
Math. J. in 2022;- the release of detailed surveys (cf. [Alien], [EssLgc]) concerning IUT;
- numerous workshops/talks/videos (cf. WS3, WS4) on IUT conducted/produced during theperiod 2012 – 2023;- a pronounced shift in recent years in the focus of activities of many researchers involved
with IUT from activities devoted to disseminating the original version of IUT to activities
centered around the development of various new versions/applications of IUT, many ofwhich are closely related to the Section Conjecture in anabelian geometry,fundamental misunderstandings and entirely unnecessary confusion concerning IUT
continue to persist in certain sectors of the mathematical community.

This includes

profoundly factually inaccurate assertions to the effect that "the experts agree that the theory
is mathematically incorrect". On the other hand, countless attempts by the author of IUT,
as well as other mathematicians involved with IUT, over the past decade --- cf., e.g., e-mails
sent on June 30, 2022, and December 30, 2022, neither of which received any response! --to pin down

- precisely who these "experts" are and/or- precisely what the mathematical content is that these "experts" assert to be "mathematicallyincorrect"
have only been met with stubborn refusals to respond to such queries. Here, it is worth
noting that this situation constitutes a very serious violation of article (6.) of the subsection
entitled "Responsibilities of authors" of the Code of Practice of the European MathematicalSociety (cf. EMSCOP):
Mathematicians should not make public claims of potential new
theorems or the resolution of particular mathematical problems
unless they are able to provide full details in a timely manner.
This confusion has also manifested itself in a sort of bizarre schizophrenia on the part of manysenior, high-ranking members of the mathematical community. That is to say,
- whereas some senior, high-ranking members of the European mathematical community
have maintained for years, both in oral communications to the author of IUT, as well as innumerous internet posts, that the incorrect mathematical content referred to above
revolves around the issue of "redundant copies" (as discussed in detail in [EssLgc]), othersenior, high-ranking members of the European mathematical community have asserted
strongly that they could not believe that this issue of "redundant copies" could be thefocal point of objections to the mathematical validity of IUT, on the grounds that they
could not believe that fellow senior, high-ranking members of the European mathematicalcommunity could be confused about such an elementary mathematical issue;- whereas some senior, high-ranking members of the European mathematical community
have maintained for years, both in oral communications to mathematicians involved with
IUT and in written form (cf., e.g., the Zentralblatt review of the IUT papers!), that "theexperts agree that the theory is mathematically incorrect", one (very) senior, high-ranking
member of the European mathematical community has asserted categorically (in apersonal oral communication) that neither he nor his colleagues take such assertions (of amathematical gap in IUT) seriously!
In light of the circumstances just described, one central aspect of dissemination activities
concerning IUT over the past decade has consisted of stimulating the worldwide mathematicalcommunity to strongly encourage mathematicians to(*) bear explicit responsibility for any assertion that denies the essential mathematical
validity of IUT by expressing any such assertion --- not in an informal manner, via roughoral communications, internet posts, or nonrigorously formulated arguments, but rather--- in the form of a precisely formulated mathematical paper with complete, rigorous
proofs that is written in such a fashion that it is suitable for publication in a wellestablished mathematical journal.
This goal (*) may be understood as the goal of achieving compliance with the passage of theEMSCOP quoted above.

Alternatively, this goal (*) may be understood as the goal ofaddressing, in a fundamental and definitive fashion, the entirely unnecessary confusion andschizophrenia discussed above, by putting assertions of the sort referred to in (*) in a form
that is well-defined and can hence be subject, via conventional scholarly activities (i.e.,workshops/seminars, further research, etc.), to the scrutiny of the mathematical community,
not only in the present, but also for decades (or even centuries!) to come (cf. the discussionof the historical significance of explicit written documentation of mathematical arguments in[EssLgc], §1.5).
Perhaps the IUT Challenger Prize announced by Mr. Nobuo Kawakami in July 2023 canfurther stimulate the worldwide mathematical community to realize the importance, for theentire field of mathematics, of conducting oneself in a fashion consistent with (*).
In some sense, the model of the many of the dissemination activities referred to above may beunderstood as the discussions between Emmanuel Lepage (a maître de conférences atSorbonne University in France) and the author of IUT during the years 2017 - 2021. At theoutset of these discussions (in the summer of 2017), Lepage took a deeply skeptical position
with regard to the mathematical validity of IUT. On the other hand, it was precisely as aresult of his sincere efforts during the period 2017 - 2021 to respond to repeated requests by
the author of IUT to make his objections to the mathematical validity of IUT precise andexplicit that Lepage was finally able to realize and acknowledge explicitly that his objections
to IUT were purely psychological, that he had misunderstood IUT, and that he no longer had
any mathematical reasons not to acknowledge the mathematical validity of IUT.

Thesediscussions with Lepage formed the basis of a substantial portion of the discussion given in
the latter portion of [EssLgc], §3. Thus, in summary, one may think of the goal of thedissemination activities referred to above as the goal of

(**) performing a massive global search for professional mathematicians who assert that they
have mathematical reasons not to acknowledge the mathematical validity of IUT, in thehope that stimulating such mathematicians to perform the "exercise" of documenting
these mathematical reasons in a precise and explicit fashion can trigger a comparably
dramatic evolution of understanding for such mathematicians to the evolution ofunderstanding experienced by Lepage during discussions with the author of IUT in
2017 - 2021.
Finally, we note that, in this context, it is of fundamental importance to distinguish between- mathematicians who do not acknowledge the mathematical validity of IUT for
nonmathematical (such as social/political/psychological) reasons and- mathematicians who do not acknowledge the mathematical validity of IUT formathematical reasons.
Indeed, one way to understand the goal of the dissemination activities referred to above is as
the goal of issuing(***) a clear and unequivocal message to the worldwide mathematical community that thedenial of the mathematical validity of a mathematical theory published in a wellestablished mathematical journal for nonmathematical reasons is entirely unacceptable(cf.,e.g., the passage of the EMSCOP quoted above!), and that

thefundamental importance, from the point of view of the sound development of the entirefield of pure mathematics, of providing mathematical reasons for any such denial of themathematical validity of a mathematical theory transcends the specific case of
IUT and indeed cuts to the very core of what it means to practice pure mathematics.