Kirchhoff’s Theorem and Semi-Canonical Representatives for the Tropical Prym Variety

Explore Kirchhoff's theorem and its applications to tropical Prym varieties in this 34-minute lecture by Dmitry Zakharov from Central Michigan University. Delve into the intricacies of metric graphs, divisors, and linear equivalence, while examining the Jacobian and Picard groups. Investigate the geometric interpretation of volume formulas and the structure of the tropical Abel-Jacobi map. Analyze double covers of graphs and metric graphs, and discover the discrete Prym for double covers of finite graphs. Gain insights into the construction of double covers, the volume of tropical Prym varieties, and the geometrization of volume formulas and Abel-Prym maps.

Intro
Kirchhoff's matrix tree theorem
Divisors and linear equivalence on metric graphs
The Jacobian and Picard groups of a metric graph
Kirchhoff's theorem for metric graphs
Geometric interpretation of the volume formula
Structure of the tropical Abel-Jacobi map
ABKS decomposition of the Jacobian
Double covers of graphs and metric graphs
The discrete Prym for double covers of finite graphs
Construction of double covers of graphs
The volume of the tropical Prym variety
Geometrization of the volume formula and Abel-Prym map