Counting open curves via Berkovich geometry

Motivated by mirror symmetry, we study the counting of open curves in log
Calabi-Yau surfaces. Although we start with a complex surface, the
counting
is achieved by applying methods from Berkovich geometry. The idea is to
relate the counting of open curves to special types of closed curves.
This
gives rise to new geometric invariants inaccessible by classical methods.
These invariants satisfy a list of very nice properties. I will talk
about
the positivity, the integrality and the gluing formula. If time
permits, I
will also mention the conjectural wall-crossing formula as well as
applications to mirror symmetry.