Amoebas, Coamoebas, and Tropical Geometry

Given a triangle with side lengths 1, log x, log y and opposite interior angles π-u-v, u, v, the functional determinant ∂(x,y)/∂(u,v) is identically equal to 1. This amusing fact is a special instance of the relation between the amoeba of a complex curve and its corresponding coamoeba. This colloquium talk is intended to give a gentle introduction to the properties and applications of mathematical amoebas. For instance, we shall explain why the spine of an amoeba is a tropical hypersurface, and what are the complex curves having amoebas of maximal area.