$A^1$ enumerative geometry: counts of rational curves in $P^2$ - 2 of 2

We will introduce $A^1$ homotopy theory, focusing on the $A^1$ degree of Morel. We then use this theory to extend classical counts of algebraic-geometric objects defined over the complex numbers to other fields. The resulting counts are valued in the Grothendieck--Witt group of bilinear forms, and weight objects using certain arithmetic and geometric properties. We will focus on an enrichment of the count of degree d rational plane curves, which is joint work with Jesse Kass, Marc Levine, and Jake Solomon.