Moduli spaces of tropical objects can often be obtained as tropicalization of suitable compactifications of algebro-geometric objects.
In the talk we will start by giving an overview of the construction of universal compactified Jacobians along with their tropical counterparts, in the category of cone stacks.
We then show that this cone stack can be obtained by tropicalizing two versions of the algebraic universal Jacobian: the logarithmic universal Jacobian and the non archimedian universal Jacobian. We then discuss, within this picture, new stability conditions for (universal) compactified Jacobians.
The talk will be based on joint work of the speaker with S. Molcho, M. Ulirsch, F. Viviani and J. Wise.