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

这个项目的目的是从动机的角度研究曲线模空间情况下的堆叠惯性。我们的框架是两重的，一方面Grothendieck-Teichmüler理论提供了一个丰富的计算双重框架，具有算术和基元方面；另一方面，最近适用于Deligne-Mumford堆栈的Morel-Voevodsky Motivic同伦理论为堆栈惯性的研究提供了一个方便的范畴。我们的研究更准确地被应用于一类中的多个Zeta值所驱动。它从考虑堆栈结构特殊性的理据范畴的定义，到可以与亏格零的理据填充-洗牌关系相比较的新的惯性关系的计算。通过Grothendieck-Teichmüler理论的两个方面，本项目扩展并依赖于最新的发展，即考虑堆栈惯性作为算术几何中的一个新的关键因素，以及同伦理论在理据理论的基础上的复兴。从SPP的角度来看，这项研究可以被视为经典算术几何和A^1-同伦理论之间的双向联系，双方都有新的发展，因此肯定会从与该计划的其他参与者的交流中受益。

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.