Abstract: Severi varieties parametrize integral curves of fixed
geometric genus in a given linear system on a surface. In this talk, I
will discuss the classical question of whether Severi varieties are
irreducible and its relation to the irreducibility of other moduli
spaces of curves. I will indicate how tropical methods can be used to
answer such irreducibility questions. The new results are from ongoing
joint work with Xiang He and Ilya Tyomkin.