Higher convexity of coamoeba complements

Frank Sottile

Amoebas and coamoebas are the images of varieties of 
the complex algebraic torus under coordinatewise 
logarithm and argument maps, respectively.  As shadows
of the original variety, they retain some of its structure.
When the variety is a hypersurface, the connected 
components of the complements of both the amoeba and 
coamoeba are convex.  Henriques introduced a homological
generalization of convexity and proved that complements 
of amoebas satisfy a weak form of this higher convexity.

   In this talk, I will explain these notions and 
describe some of the structure of coamoebas, 
namely their phase limit sets and shells, and then 
sketch how to use this structure to show that complements
of coamoebas have this higher convexity of Henriques.
This is joint work with Mounir Nisse.