Caelan Ritter (UW)

PDL C-38

**Pre seminar:**An introduction to tropicalization

We will discuss various ways to go from the algebraic world to the tropical world (and why one might want to do such a thing), with an emphasis on degenerations of families of curves over non-archimedean valuation rings.

**Seminar:**A graph invariant from tropical homology and the Ceresa cycle

The Jacobian of a very general complex algebraic curve of genus at least 3 contains an algebraic cycle called the Ceresa cycle that is homologically trivial but algebraically nontrivial. An analogous notion due to Zharkov exists in the tropical setting, with a similar algebraic nontriviality statement for metric graphs overlying the complete graph on four vertices. We extend this result by considering a related, "universal" invariant of the underlying graph; we show that triviality of this invariant has a forbidden minor characterization that coincides with the graph being of hyperelliptic type.