Motivic homotopy theory and stable homotopy groups

TL;DR

This paper surveys motivic homotopy theory over a field, focusing on computations of motivic homotopy sheaves and applications to stable stems via the motivic Adams spectral sequence, linking to synthetic homotopy theory.

Contribution

It provides a comprehensive overview of motivic homotopy theory, including recent computational techniques and their implications for stable homotopy groups.

Findings

Computed motivic homotopy sheaves both stably and unstably.

Applied motivic Adams spectral sequence to determine stable stems.

Connected motivic homotopy theory with synthetic homotopy theory.

Abstract

These notes, written version of a Bourbaki talk, survey Morel-Voevodsky's motivic homotopy theory over a field, with a focus on computations of motivic homotopy sheaves, both stable and unstable. We also describe Isaksen-Wang-Xu's applications to the determination of stable stems through the motivic Adams spectral sequence, which also paved the way toward synthetic homotopy theory.