Tropicalization of Classical Moduli Spaces

Algebraic geometry is the study of solutions sets to polynomial equations.
Solutions that depend on an infinitesimal parameter are studied combinatorially by
tropical geometry. Tropicalization works especially well for varieties that are
parametrized by monomials in linear forms. Many classical moduli spaces (for curves
of low genus and few points in the plane) admit such a representation, and we here
explore their tropical geometry. Examples to be discussed include the Segre cubic,
the Igusa quartic, the Burkhardt quartic, and moduli spaces of marked del Pezzo
surfaces. Matroids, hyperplane arrangements, and Weyl groups play a prominent role.
Our favorites are E6, E7 and G32.  This is joint work with Qingchun Ren and Steven Sam.