The tropical vertex group, scattering diagrams and quivers

The tropical vertex group $V$ introduced by Kontsevich-Soibelman plays a role in many problems in algebraic geometry and mathematical physics. The group itself can be understood in very different ways. In the approach due to Gross, Pandharipande and Siebert, a central role is played by tropical curves in $\mathbb{R}^{2}$ and their enumerative invariants. This approach leads to a number of applications. First we will describe the tropical vertex group and in particular scattering diagrams. Then we will sketch the connection with tropical curves and, if time permits, with moduli spaces of quiver representations.