Report on the Oxford workshop on IUT theory of Shinichi Mochizuki  by Ivan Fesenko

The inter-universal Teichmüller (IUT) theory of Shinichi Mochizuki operates with full Galois and fundamental groups, which already makes it different from class field theories, Langlands correspondences and higher class field theories. Its main object - theatres is a non-scheme theoretical extension of usual scheme theoretical objects such as the idele class group. Non-scheme theoretical arithmetic deformation tools are applied to the theatres, they are called theta-links and log-links. Key words include arithmetic fundamental groups, mono-anabelian reconstruction of number and local fields using hyperbolic curves over number fields, theta function, line bundles and theta values on Tate's curve and rigidities of mono-theta-environment, generalised Kummer map, Galois evaluation, bridges between arithmetic and geometry, mono-anabelian transport, multiradiality as a monoid-categorical generalisation of functoriality, multiradiality and indeterminacies, theatres and theta-link, log cores, log-theta-lattice, arithmetic deformation. For introductory texts see this page as well as a new survey of IUT by its author and this very general introductory text which appeared after the Oxford workshop.

The inter-universal Teichmüller theory, also known as arithmetic deformation theory, is based on several new concepts and insights. To study and understand IUT, one has to invest an appropriate amount of time in its study, at least several months of continuous efforts.

The Oxford workshop was the first international workshop on IUT theory of Shinichi Mochizuki. It aimed to assist mathematicians interested to study IUT. More than 60 registered participants of the workshop included 14 master and PhD students students; areas included number theory, logic and geometry. The organisers had received various requests to video record the talks of the workshop from mathematicians unable to attend and were already close to making arrangements for this and pay the cost from other sources of funding, but this initiative was blocked.

Prior to the workshop its participants had been advised to read a number of texts, including some texts on anabelian geometry, which is a compulsory prerequisite for IUT. Around 50 questions to the author of the theory from the participants had been received and answered by the author of the theory before the workshop. 

The program of the workshop covered the main prerequisite papers, i.e. papers of Shinichi Mochizuki preceding IUT papers and used in IUT theory, published in the last twenty years, their approximate volume is 1300 pages. These talks on the prerequisite papers did not cause essential problems in their understanding. The first day talks included talks on more standard topics such as a proof of geometric Szpiro inequality inspired by Bogomolov (Zhang), various versions of the Belyi maps and their use in the reduction of the Vojta conjecture (Kühn), and a reduction of the Vojta conjecture which is used in the IUT papers (Javanpeykar). The first day also included a very useful introduction to IUT and presentation of mono-anabelian transport (Hoshi) which plays a central role in IUT. The second day was dedicated to absolute anabelian geometry (Tan, Stix, Kuhne). The third day dealt with categorical geometries used in IUT such as frobenioids (Ben-Bassat, Czerniawska) and semi-graph anabeliods (Szamuely, Lepage). A talk about Hodge-Arakelov theory (Yamashita) explained some of motivation for several concepts of IUT. Two talks on the etale theta function (Kedlaya) reviewed the use of the nonarchimedean theta function, its special values and associated rigidities discovered by Mochizuki.

The IUT papers, their volume is approximately 500 pages, were sketched, in some general form, during the last two days. Talks by Mok and Hoshi on the fourth and fifth days of the workshop gave certain glimpses of the theory. As expected, these relatively short talks on the first two IUT papers were quite difficult to deliver as well as to follow. Animations illustrating one of the main theorems of IUT-III were produced prior to the workshop and their meaning was explained during the workshop.

Several of the main concepts and objects (listed in the letter to participants) were either presented at the workshop or are discussed in slides and notes of the workshop. The central place in the program of the Oxford workshop was occupied by two skype sessions of questions and answers with Shinichi Mochizuki, two hours long each. All asked questions were given comprehensive answers.

Substantial efforts were applied to find speakers for the workshop. Younger participants volunteered to give many talks. Most of the speakers can be commended for their efforts to prepare and deliver their talks. A larger than usual at mathematical conferences number of rushed and irrelevant questions was asked by a very small number of participants who declined to follow the organisers' request to prepare and deliver talks at the workshop. One of such questions was why one needs to use categories to prove the abc conjecture: the author of that question managed to miss in his education a huge part of arithmetic geometry such as Deligne's proof of the GRH in positive characteristic and a substantial body of Grothendieck's work. Relatively high numbers of mathematically weak questions were asked by another participant who despite his demonstrated during the workshop lack of grasp of the subject matter found appropriate to put online his personal notes containing many mathematically incorrect statements.

The workshop helped its participants to go through the prerequisites of the theory and to see many main new concepts of the theory in action, so that with appropriate self-work these mathematicians can be more successful in their study of IUT. It was interesting to observe that some participating geometers and logicians, as well as several young researchers were progressing faster in their understanding of new concepts of IUT than other participants. Several logicians made very interesting remarks on some of the concepts of IUT. The workshop concluded with a general feeling of optimism. Extended workshop proceedings, which also include some additional material, are available from this page, in particular they contain more information about the structure of the program of the workshop. The subsequent RIMS workshop built on the materials the Oxford workshop, thus dedicating most of its time to the IUT papers. 15 of the participants of the Oxford workshop attended the RIMS workshop and 7 of its speakers gave talks there.