applications of the tropical vertex group

The tropical vertex group is generated by certain formal symplectomorphisms of the 2-dimensional algebraic torus. It plays a role in some problems in algebraic geometry and mathematical physics, e.g. wall-crossing. It is known that the group itself can be understood in many ways, for example in terms of "counting" certain rational curves or representations of quivers. This leads to nice correspondences. I will discuss joint work with M. Reineke and T. Weist in which we determine the "dual" of a remarkable formula of Manschot, Pioline and Sen for quiver representations: it is a degeneration formula in Gromov-Witten theory. If time permits I will also mention joint work in progress with S. A.
Filippini in which we study the tropical counts underlying the group using an integral equation of Gaiotto, Moore and Neitzke.