Tropical-Toric connections and applications to Wonderful compactifications

Tropical geometry has a nice connection with toric intersection theory. More precisely, certain intersection numbers that appear in toric intersection theory can be interpreted as multiplicities of certain tropical varieties. In this talk, I will explain this connection more in detail, and show how to get an algorithm out of it that allows us to compute the intersection class of a subvariety of a toric variety from the data of its tropicalization. Finally, I will discuss some applications and open problems on Wonderful Compactifications.