  We review the recent proof of the N.Takahashi's conjecture on genus $0$
Gromov-Witten invariants of $(\mathbb{P}^2, E)$, where $E$ is a smooth cubic
curve in the complex projective plane $\mathbb{P}^2$. The main idea is the use
of the algebraic notion of scattering diagram as a bridge between the world of
Gromov-Witten invariants of $(\mathbb{P}^2, E)$ and the world of moduli spaces
of coherent sheaves on $\mathbb{P}^2$. Using this bridge, the N.Takahashi's
conjecture can be translated into a manageable question about moduli spaces of
coherent sheaves on $\mathbb{P}^2$.
  This survey is based on a three hours lecture series given as part of the
Beijing-Zurich moduli workshop in Beijing, 9-12 September 2019.
