Ralph Morrison: Tropical Geometry Mini-course, Part II: Tropicalizations and the Fundamental Theorem

In the first talk of this mini-course, we defined tropical varieties in terms of polynomials over a min-plus algebra.  In this talk we offer an alternative definition of tropical geometry that connects it more directly to algebraic geometry.  The tropicalization map sends a variety over a non-Archimedean field into Euclidean space under coordinate-wise valuation. The Euclidean closure of the image is called a tropical variety.  We will see through the Fundamental Theorem of Tropical Geometry that these two characterizations agree.  We will close by looking at how to lift information from the tropical world to the realm of algebraic geometry.