Motivic Stack Inertia of Moduli Spaces of Curves, Variation of Periods and Multiple Zeta Values in Genus 0 and 1

• Applicant: Benjamin Collas, Ph.D.
• Subject Area: Mathematics
• Term: from 2015 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 269705732

The goal of this project is to study the stack inertia in the case of moduli spaces of curves from a motivic point of view. Our framework is two folds, with on one hand the Grothendieck-Teichmüller theory providing a rich and computational dual-framework with arithmetic and motivic sides; and on other hand the recent Morel-Voevodsky motivic homotopy theory adapted to Deligne-Mumford stacks providing a convenient category for the study of stack inertia. Our study is more precisely driven by applications to Multiple Zeta Values in genus one. It goes from the definition of a motivic category that takes into account the specificities of the stack structure, to the computation of new inertia relations that can be compared with the motivic stuffle-shuffle relation from genus zero.Through the two sides of Grothendieck-Teichmüller theory, this project extends and relies on very recent developments which are the consideration of stack inertia as a new key ingredient in Arithmetic Geometry, and the revival of Homotopy Theory in the foundation of Motivic theory. From the point of view of this SPP, this study can been seen as a two ways connection between classical Arithmetic Geometry and A^1-homotopy theory with new developments in both sides, and as such will certainly benefit from exchanges with other participants from the Programme.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1786: Homotopy Theory and Algebraic Geometry
• Co-Investigator: Professor Dr. Michael Dettweiler