### Multiple end solutions to the Allen-Cahn equation in ${\mathbb{R}}^{2}$

An entire solution of the Allen-Cahn equation $\Delta u=f\left(u\right)$, where $f$ is an odd function and has exactly three zeros at $\pm 1$ and $0$, e.g. $f\left(u\right)=u({u}^{2}-1)$, is called a $2k$ end solution if its nodal set is asymptotic to $2k$ half lines, and if along each of these half lines the function $u$ looks (up to a multiplication by $-1$) like the one dimensional, odd, heteroclinic solution $H$, of ${H}^{\text{'}\text{'}}=f\left(H\right)$. In this paper we present some recent advances in the theory of the multiple end solutions. We begin with the description of the moduli space of such solutions....