Combinatorial Algebraic Geometry: Tropical Ceresa cycles

The Ceresa cycle C - C^{-} of a smooth algebraic curve C is a tautological algebraic cycle contained in the Jacobian J(C).  It is homologically trivial, but Ceresa showed that if C is very general of genus at least 3, then it is not algebraically trivial.  It is in some sense the simplest algebraic cycle satisfying these properties, leading to applications for the étale fundamental group, intersection theory, and the theory of heights.  We will discuss the extent to which the Ceresa cycle and the proof of non-triviality carry over to the tropical setting.  Along the way, we will introduce important tools in the study of rational polyhedral spaces, namely, tropical cycles and homology.