The Structure of Motivic Homotopy Groups

We study the stable motivic homotopy groups $\pi_{s,w}$ of the 2-completion of the motivic sphere spectrum over $\mathbb{C}$. When arranged in the $(s,w)$-plane, these groups break into four different regions: a vanishing region, an $\eta$-local region that is entirely known, a $\tau$-local region that is identical to classical stable homotopy groups, and a region that is not well-understood.