Inspired by Morel's \(\mathbb{A}^1\)-degree which takes values in the Grothendieck-Witt ring \(GW(k)\) of a field \(k\), Kass and Wickelgren define the \(\mathbb{A}^1\)-Euler number to be the some of local \(\mathbb{A}^1\)-degrees of the zeros of a general section of a vector bundle \(\pi:E\rightarrow X\) of rank equal to the dimension of \(X\). As an application they count the lines on cubic surfaces, that is the \(\mathbb{A}^1\)-Euler number of \(Sym^3\mathcal{S}^*\rightarrow Gr(2,4)\)where \(\mathcal{S}\rightarrow Gr(2,4)\) is the tautological bundle over the Grassmanian of lines in \(\mathbb{P}^3\), to be \(15\langle1\rangle+12\langle-1\rangle\in GW(k)\) which recovers the classical complex (27=rank) and real (3=signature) count.
In my talk I will give an introduction to \(\mathbb{A}^1\)-enumerative geometry via the \(\mathbb{A}^1\)-degree and will compute several \(\mathbb{A}^1\)-Euler numbers.