EXTENDED PROCEEDINGS OF

Oxford (CMI and C&S) workshop on IUT theory of Shinichi Mochizuki

December 7-11 2015, L3 Math Institute Univ. Oxford

I. Fesenko (scient. organizer and editor)


Letter to participants (I. Fesenko)


Photos of the speakers


The content of papers may differ from the content of talks




December 7


9:00-9:20 Ivan Fesenko Foreword 0

9:20-10:10 Shou-Wu Zhang A proof of geometric Szpiro inequality inspired by Bogomolov 1

10:40-11:40 Ulf Kühn Arithmetic of elliptic curves in general position 2

13:30-14:30 Ariyan Javanpeykar Reducing the conjectures to one theorem of IUT 3

15:00-16:00 Yuichiro Hoshi An approximate statement of the main theorem of IUT 4

16:15-17:15 Yuichiro Hoshi Mono-anabelian transport 4

17:20-18:00 Discussions



December 8


9:00-11:00 Skype session with Shinichi Mochizuki

11:30-12:30 Fucheng Tan Absolute anabelian geometry 5

13:50-14:50 Fucheng Tan Absolute anabelian geometry 5

15:20-16:20 Jakob Stix Reconstruction of fields using Belyi cuspidalization 6

16:35-17:35 Lars Kuehne Archimedean aspects of absolute anabelian geometry 7

17:35-18:00 Discussions



December 9


9:00-9:45 Oren Ben-Bassat Frobenioids 1 8

10:00-10:45 Weronika Czerniawska Frobenioids 2 9

11:15-12:15 Tamás Szamuely Geometry and semi-graphs of anabelioids 10

14:10-15:10 Emmanuel Lepage Tempered fundamental group of semi-graphs of anabelioids 11

15:45-16:15 Go Yamashita Motivation from Hodge-Arakelov theory 12

16:30-17:30 Kiran Kedlaya Etale theta function 13

17:30-18:00 Discussions



December 10


9:00-10:00 Kiran Kedlaya Etale theta function 13

10:15-11:15 Chung Pang Mok IUT, Introduction to (Hodge) theatres 14

11:45-12:45 Chung Pang Mok IUT, Introduction to (Hodge) theatres 14

13:50-14:50 Chung Pang Mok IUT, Multiradiality of the theta environment 15

15:20-16:20 Yuichiro Hoshi IUT, Hodge-Arakelov-theoretic evaluation 16

16:35-17:35 Yuichiro Hoshi IUT, Hodge-Arakelov-theoretic evaluation 16

17:35-18:00 Discussions



December 11


9:00-10:00 Go Yamashita IUT, [IUT-III] 17

10:15-11:15 Go Yamashita IUT, [IUT-III] 17

11:45-12:45 Go Yamashita IUT, [IUT-III] 17

13:40-15:40 Skype session with Shinichi Mochizuki 18

16:10-17:10 Go Yamashita IUT, [IUT-III-IV], with some remarks on the language of species 19

17:10-17:30 Ivan Fesenko Summary and what's next






   Talks on Monday included "classical topics" talks (1-3) and two talks (4) aimed at giving the first glimpse of IUT.
   Talks on Tuesday dealt with anabelian, absolute and semi-absolute anabelian and mono-anabelian issues.
   The first four talks on Wednesday dealt with categorical geometry useful for IUT.
   The etale theta function paper is a prerequisite for IUT.
   The remaining talks on Thursday and Friday were aimed to introduce, using three different styles of presentation, some of key features of IUT.
   A detailed presentation of IUT papers will be arranged during the next workshop in Kyoto.


0 please see this page for survey texts of IUT which you may find useful to read first

1 this proof can be viewed as an elementary guide or blueprint for IUT, see [B] and also this table; the key inequality is actually the Milnor-Wood inequality, see e.g. this paper

2 [NB] and [AE]

3 Sect. 1-2 of [IUT-IV] without proof of Th 1.10 of [IUT-IV]

4 these follow a very recent survey of IUT by Yuichiro Hoshi, in Japanese.
        One can also recommend three slide talks of the same speaker at a RIMS conference in December 2015 available from section Lectures of this page

5 [TAAG-I-II]

6 sect.1 of [TAAG-III]

7 sect. 2 and 4 of [TAAG-III]

8 [F], see also Responses to questions on Frobenioids by Shinichi Mochizuki, in particular Response 8

9 [F], see also Responses to questions on Frobenioids by Shinichi Mochizuki, in particular Response 8

10 [A]

11 [A] and sect. 2 of [IUT-I]

12 [HAT], notes on Hodge-Arakelov theory by Robert Kucharczyk prepared for his cancelled talk and slides 12-23 of a talk by Go Yamashita at RIMS in March 2015

13 [ET], picture of covers

14 [IUT-I]

15 sect. 1 of [IUT-II]

16 sect. 2-4 of [IUT-II]

17 [IUT-III]

18 see in particular Response on a question of Fucheng Tan by Shinichi Mochizuki

19 [IUT-III] and sect. 1 of [IUT-IV], with some remarks on the language of species, sect. 3 of [IUT-IV]




All papers below are authored by Shinichi Mochizuki and available, often with comments, from this page


[A] The geometry of anabelioids, Publ. Res. Inst. Math. Sci. 40 (2004), 819–881;

Semi-graphs of anabelioids, Publ. Res. Inst. Math. Sci. 42 (2006), 221–322

[AE] Arithmetic elliptic curves in general position, Math. J. Okayama Univ. 52 (2010), 1–28

[B] Bogomolov's proof of the geometric version of the Szpiro conjecture from the point of view of inter-universal Teichmüller theory,
preprint 2015

[ET] The étale theta function and its frobenioid-theoretic manifestations, Publ. Res. Inst. Math. Sci. 45 (2009), 227–349

[F] The geometry of frobenioids I: The general theory, Kyushu J. Math. 62(2008), 293–400;

The geometry of frobenioids II: Poly-Frobenioids, Kyushu J. Math. 62 (2008), 401–460

[HAT] A survey of the Hodge–Arakelov theory of elliptic curves I, in Proc. of Symp. Pure Math. 70, AMS (2002), 533–569;

A survey of the Hodge–Arakelov theory of elliptic curves II, Adv. Stud. Pure Math. 36, Math. Soc. Japan (2002), 81–114

[IUT] Inter-universal Teichmüller theory I: Constructions of Hodge theaters, preprint 2012–2015;

Inter-universal Teichmüller theory II: Hodge-Arakelov-theoretic evaluation, preprint 2012–2015;

Inter-universal Teichmüller theory III: Canonical splittings of the log-theta-lattice, preprint 2012–2015;

Inter-universal Teichmüller theory IV: Log-volume computations and set-theoretic foundations, preprint 2012–2015

[NB] Noncritical Belyi maps, Math. J. Okayama Univ. 46 (2004), 105–113

[TAAG] Topics in absolute anabelian geometry I: Generalities, J. Math. Sci. Univ. Tokyo 19 (2012), 139–242;

Topics in absolute anabelian geometry II: Decomposition groups and endomorphisms, J. Math. Sci. Univ. Tokyo 20 (2013), 171–269;

Topics in absolute anabelian geometry III: Global reconstruction algorithms, J. Math. Sci. Univ. Tokyo 22 (2015), 939-1156