  In this thesis we study the relation between scattering diagrams and
deformations of holomorphic pairs, building on a recent work of Chan--Conan
Leung--Ma. The new feature is the extended tropical vertex group where the
scattering diagrams are defined. In addition, the extended tropical vertex
provides interesting applications: on one hand we get a geometric
interpretation of the wall-crossing formulas for coupled $2d$-$4d$ systems,
previously introduced by Gaiotto--Moore--Neitzke. On the other hand,
Gromov--Witten invariants of toric surfaces relative to their boundary divisor
appear in the commutator formulas, along with certain absolute invariants due
to Gross--Pandharipande--Siebert, which suggests a possible connection to
open/closed theories in geometry and mathematical physics.
