## Arithmetic Homotopy Geometry

**Arithmetic Homotopy Geometry** at RIMS -- seminars, workshops, talks -- in collaboration and with the support of my RIMS colleagues (esp. Prof. Tamagawa, Mochizuki, Hoshi, Tsujimura, Minamide, Sawada).

Personal interest in the moduli stacks of curves and its stack inertia stratification — see for a glimpse; Includes simplical and homotopical algebraic geometry — i.e. model categories and motivic considerations for stacks à la *Morel-Voevodsky* —, the *arithmetic geometry of curves* — e.g. G-covers, irreducible components of Hurwitz spaces, étale fundamental group —, *Grothendieck-Teichmüller theory* — e.g. mapping class groups, pants decompositions, Serre bonté —, the *arithmetic of operads* — in genus 0 via Friedlander étale topological type and prospaces, and the *Tannakian formalism in Perverse sheaves*.

Since 2023 this activity takes place in the CNRS France-Japan International Research Network LPP-RIMS ``Arithmetic & Homotopic Galois Theory'' (AHGT), see seminar and workshops, and publications.

### Arithmetic Homotopy Geometry - Seminars and Workshops

The *Arithmetic and Homotopic Galois Theory Seminar* (form. *RIMS - Homotopic Arithmetic Geometry Seminars Series*) introduces key results of classical arithmetic geometry in relation with recently developed techniques and principles (incl. a special session ``Promenade in Inter-universal Teichmüller theory'' with Lille University, France).

The *Oberwolfach workshops series (2018, 2021, 2023)* presents the latest international developments of the field and investigates new research directions.

## Recent Events

### RIMS - Expanding Horizons of Inter-universal Teichmüller Theory 2021-2022

Inter-universal Teichmüller theory can be seen as a new kind of geometry, which beyond Grothendieck's ring-scheme algebraic geometry, reunites seminal anabelian, arithmetic and Diophantine insights. The goal of this series of workshops (WS1-4) is to present techniques and principles of these theories in relation with the absolute Galois group of the rational numbers (and its GT combinatoric variant).

We refer to the official page of Prof. S.Mochizuki's project for further details on programme, dates, and participants of the Workshops WS1-WS4.

## Selected Talks

- DMV Annual Meeting 2020,
*``Differential and Hodge Theoretic Methods in Algebraic Geometry'' Minisymposium*, in Chemnitz, Germany (09.2020); - IInd Kyoto-Hefei Workshop on Arithmetic Geometry 2020, China (08.2020);
- 5th KTGU Mathematics, at RIMS Kyoto, Japan (02.07);
- Algebra Seminar, at Hiroshima University, Japan (01.20);
- Algebraic Number Theory and Related Topics, at RIMS Kyoto, Japan (12.19);
- Number Theory Seminar, at Kyoto University, Japan (11.19);
*Moduli stacks of curves: arithmetics and motives*, at NTNU, Norway (08.19);*Galois Arithmetic of Perverse Sheaves in genus 1*, at XXIXèmes Rencontres Arithmétiques de Caen: "Field Arithmetic and Arithmetic Geometry" (06.19);*"Landweber Exact Functor Theorem: MU, Stack inertia & flatness''*& org. of the Bayerische AG "Towards Chromatic Homotopy Theory" (03.19).*Arithmetic of Operads of Moduli Spaces of Curves*, at Temple University, US (03.18);

## Selected Activity

### Homotopical Arithmetic Geometry of Stacks - RIMS 2018

Moduli stacks of curves are ideal spaces in the study of geometric Galois representations and motivic theory. In these series of lectures, we present fundamental geometric and arithmetic properties of these spaces -- *I & II - Algebraic & Deligne-Mumford Stacks, III - Moduli Problems & Moduli Spaces of Curves, IV - Fundamental Group & Arithmetic, V - Motivic Theory for Moduli Stack of Curves* -- the goal being to introduce some elements of arithmetic homotopy theory (etale homotopy type, homotopical stacks, Morel-Voevodsky motivic homotopy).

*For details see RIMS - Number Theory / Arithmetic Geometry Seminar and Programme for references.*