Diane Maclagan: Tropical schemes, tropical cycles, and valuated matroids

The tropicalization of a subvariety of a torus records the cycle of its compactification in an ambient toric variety.  In a recent preprint Jeff and Noah Giansiracusa introduced a notion of scheme structure for tropical varieties, and showed that the tropical variety as a set is determined by this tropical scheme structure.  I will introduce these notions, and outline how to also recover the tropical cycle from this information.  The lurking combinatorics is that of valuated matroids. This is joint work with Felipe Rincon.