The ACO Seminar (2022–2023)

A graph can be viewed, in many respects, as an analogue of a Riemann surface. For example, any graph comes with a Jacobian which is a real torus and is closely related to the celebrated matrix-tree theorem. There is also an Abel-Jacobi theory, providing maps from a graph to its Jacobian, so one can draw a graph in its Jacobian. This picture gives rise to various interesting canonical measures on graphs, some of which related to the theory of random spanning trees.

This talk will be self-contained and aimed at a general mathematical audience. Somewhat surprisingly, these ideas have origins and applications in arithmetic and algebraic geometry, but we will not delve into those connections in this talk.