TOP > Members > COLLAS, Benjamin

Research Field

My research focuses on developing the Arithmetic Homotopy Geometry of the moduli stacks of curves, mainly in terms of their stack inertia stratification — see for a glimpse. My work 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.

RIMS - Expanding Horizons of Inter-universal Teichmüller Theory 2021

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.

RIMS - Homotopic Arithmetic Geometry Seminars 2019 -...

The goal of the Homotopic Arithmetic Geometry Seminars Series is to introduces key results of classical arithmetic geometry in relation with simplicial homotopy theory. Semester '20-21 is a special session with Lille University (France) on IUT geometry.

See the page of each semester below for programmes, references, and list of participants. See also the Oberwolfach workshops 2021 and 2018 on Homotopic Arithmetic Geometry and Galois theory.

Selected Talks

Selected Activity

Homotopic and Geometric Galois Theory - Oberwolfach 2021

A fundamental idea in studying the absolute Galois group of a field is to make it act on geometric objects such as Galois covers, étale cohomology groups and fundamental groups. Striking advances have recently shed new light on the seminal topics of (a) Galois Covers, (b) Motivic Representations, and (c) Anabelian Geometry. Essential crossbridging principles connect these advances: homotopic methods, higher stacks, Tannakian symmetries. Based on the recent results and their promising connections, and on the 2018 MFO mini-workshop in a similar spirit, this workshop aims to crystallize these innovative approaches and to strengthen fruitful desire paths in homotopic and geometric Galois theory.
With P. Dèbes, H. Nakamura, and J. Stix - see Report of the workshop (82 pages), and previous opus in 2018.

Promenade in Inter-Universal Teichmüller Theory - RIMS-Lille 2020-21

Mochizuki's Inter-Universal Teichmüller theory provides a new geometry of Diophantine properties of the moduli stack of elliptic curves based on a deconstruction-reconstruction process -- or Fukugen 復元 -- of absolute mono-anabelian geometry. The goal of this seminar between Painlevé-Lille (France) and RIMS-Kyoto (Japan) is for generic arithmetic-geometers to gain a general understanding of results, insights and techniques of IUT.
See the webpage of the seminar for programme, schedule, participants and references.

Towards Chromatic Homotopy Theory - Bayerische AG 2019

The Landweber Exact Functor Theorem (LEFT) stands at the intersection of algebraic topology (spectra & cohomology theories), algebraic geometry (moduli stack, elliptic curves), and number theory (class field theories); it provides a general understanding of how ordinary cohomology, complex K-theory and Thom cobordism theories are related to isomorphism classes of formal group laws, and is a keystone in the potential description of the homotopy group of spheres via modular forms. This BAG presents a modern formulation of the LEFT in terms of stack and spectra for algebraic topologists and geometers.
With T. Keller and E. Koeck - see Programme for details and references.

Selected Lecture

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.



Research Institute for Mathematical Sciences (RIMS)