Brief report on Inter-universal Teichmüller Theory Summit 2016 (RIMS workshop), July 18-27 2016 

This was a fruitful and rewarding workshop on the IUT theory. The workshop was a remarkable opportunity for mathematicians from all over the world to learn many details of inter-universal Teichmüller theory (IUT) developed by Shinichi Mochizuki in his home institution.

The workshop was preceded by the Oxford IUT workshop held in December 2015. 15 of the Oxford workshop participants attended the RIMS workshop, and 7 of the Oxford workshop speakers spoke at the RIMS workshop. The materials of the Oxford workshop were of help to the participants of the RIMS workshop in the sense that reading these materials enabled to go relatively fast through several prerequisites topics.

There were 56 registered participants of the workshop. Participants came from Japan, the UK, the USA, China, Poland, France. Several Kyoto mathematicians attended some of the talks without registration. The participants’ research interests varied, to include arithmetic geometry, Diophantine geometry, anabelian geometry, the Langlands program, algebraic geometry, categorical geometry, analytic number theory, higher class field theory, zeta functions, representation theory, Iwasawa theory, dynamical systems, mathematical physics, homotopy theory and its applications, and many other areas.

Prior to the workshop, the organisers prepared a document “On Questions and Comments Concerning Inter-universal Teichmüller Theory (IUT)”.

42 talks were delivered during the 8 days of the workshop by 20 speakers. The workshop was characterised by an overall friendly atmosphere and a very concentrated work by its speakers and participants. The success of the workshop is directly related to a very substantial amount of time invested by its speakers in preparation of their talks.

The author of the theory delivered 5 hours of talks in one day, and overall 8 hours of talks. A new survey of the theory was completed by him prior to the workshop. He also conducted 8 hours of discussion sessions, answering many types of questions on many aspects of the theory. To some degree, one can say that during the workshop the author of the theory climbed several times the summit of IUT helping participants of the workshop to substantially improve their understanding of the theory. A bouquet of flowers was presented to the author of IUT on behalf of its participants on the last day of the workshop.

Many speakers gave very well prepared and sometimes outstanding talks, uncovering the beauty and  strength of the IUT theory of Shinichi Mochizuki. Younger speakers covered in their talks almost everything except IUT-III-IV. In particular, Arata Minamide was a brilliant presenter in his three lectures. He displayed a very thorough understanding of IUT which was obtained in just a few months, essentially by studying on his own. He has contributed yet another good example of a graduate student mastering, in a rather short period of time, a substantial portion of the theory that some renowned number theorists find difficult to study. Talks on IUT-III-IV were given by Yuichiro Hoshi, who had written two long surveys of IUT prior to the workshop, in Japanese, and by Go Yamashita.

Two days of the workshop included talks on relations between IUT and other areas and on potential applications of IUT. Kobi Kremnitzer talked about an arithmetic version of bounded cohomology and its potential link with IUT. Boris Zilber talked about a model theoretical approach to anabelian geometry and the Grothendieck section conjecture. Paul Vojta talked about his new version of the Siegel-Thue method. Ivan Fesenko talked about IUT in relation to class field theory and 2-dimensional adelic analysis and geometry, his talk reviewed some of the results in those theories from the point of view of IUT and proposed several open problems. Vesselin Dimitrov talked about potential applications of IUT, which use the main theorems of IUT in IUT-III-IV and some standard (for arithmetic geometers) effectivity arguments. There were many other interesting talks.

Asking concrete questions is well known to be the best way to learn new theories and this is especially so for IUT. Ample opportunities to ask concrete questions were available during the workshop. Those who intensively used the opportunity to ask questions visibly progressed in their understanding of IUT. Several open problems related to IUT were discussed during the workshop. Young researchers are welcome to start to work on them.

Files of slides of many talks and photos of the workshop are available from subpages of the workshop webpage. Materials of the workshop can be used by individual mathematicians and groups of mathematicians to study IUT.

Mathematicians willing to watch video records of the talks can contact one of the organisers of the workshop to arrange their viewing in his home department.