Gromov-Hausdorff limits of curves with flat metrics and non-Archimedean geometry

The subject of this talk is the study of the Gromov-Hausdorff limit of
a family of complex curves over a punctured disc with maximal
unipotent monodromy endowed with normalized flat metrics with conical
singularities. The limit turns out to be a metric graph which can be
naturally identified with a quotient of a subset of the Berkovich
analytic space associated to the family. This problem is inspired by
the approach of Kontsevich and Soibelman to the SYZ conjecture, and,
time permitting, I will discuss how the techniques of the talk can be
extended to be applied in this more general setting.