Tropical and Berkovich spaces

    Tropical varieties are piecewise linear shadows of algebraic varieties
with fixed embedding. They enjoy a fruitful interaction between geometry and
piecewise linear combinatorics. On one side, we can recover geometric
information, such as Hodge numbers, from tropical variety. On the other
side, classical geometric problems have (yet unsolved) tropical counterparts.
Tropical varieties are not functorial (due to presence of embedding). This
deficiency is removed when we pass to Berkovich spaces, which are limits of
tropical varieties and at the same time possess an intrinsic interpretation.
   

My talk would be an introduction to tropical varieties (mostly) and
Berkovich spaces. I will try to give an overview, concentrating on examples of
fruitful interactions with geometry. No specific knowledge from algebraic geometry
(let alone tropical geometry) is assumed.