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

The inter-universal Teichmüller (IUT) theory of Shinichi Mochizuki is not only a program which opens a new branch of number theory but also its realization via its applications to several most famous problems. IUT operates with full Galois and fundamental groups unlike class field theories, Langlands correspondences and higher class field theories. Its objects such as theatres extend scheme theoretical objects outside conventional arithmetic geometry, so that arithmetic deformation tools become available. 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 many radically new concepts and insights. To understand IUT, one has to invest an appropriate amount of time in its study, whatever one’s previous knowledge is. This study requires substantial efforts, somehow similar to efforts applied by best PhD students during their PhD years in new for them areas. In addition to attending talks at conferences, a significant amount of time, several hundred hours, must be dedicated to self-study of the theory.

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. 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. Most of the talks were generally well prepared. 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 an 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 only briefly sketched during the last two days. Three hours of talks by Mok, two hours of talks by Hoshi, and four hours of talks of Yamashita on the fourth and fifth days of the workshop included three different styles of presentation of some of the key concepts and objects of the theory. 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. As expected, the relatively short talks on the IUT papers were more difficult to deliver, as well as to follow for the participants. Animations illustrating one of the main theorems of IUT-III were produced prior to the workshop and their meaning was explained during 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. Overall, more than 300 questions were asked and answered during the workshop. Yuichiro Hoshi and Go Yamashita sometimes made comments or corrected speakers. The speakers can be commended for their substantial time and effort investment to prepare and deliver their talks. The total investment of time of the organisers and speakers of the workshop to prepare it was perhaps more than 2000 hours.

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.

Some senior funded participations had been requested to prepare talks on relatively easy topics but declined, and then younger non-funded participants, including PhD students, replaced them as speakers. The learning progress during the workshop often correlated with the prior preparation for the workshop. Participants who came to the workshop without preceding serious study, in the hope to catch everything during the talks of the workshop, often did not progress well. Few participants, who came relatively unprepared and declined to prepare talks, kept asking during the workshop some rushed, irrelevant or trivial questions, thus disturbing the flow of some of the talks. Some of the speakers complained about one of such interruptors, who later put online a very shallow post containing dozens of mistakes. In contrast, several hard-working participants, who later continued to the RIMS workshop, did not engage with journalists and did not produce any online texts.

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. 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 and used materials of its first three days, 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.