Ribbon Graphs and Mirror Symmetry

Starting from a ribbon graph, I will
define a category which serves as a stand-in
for the Fukaya category of the associated punctured
Riemann surface, thought of as a large-volume
limit.  When the ribbon graph has a combinatorial
version of a torus fibration with section, a
mirror "large complex" limit exists,
a singular algebraic curve.  In this
case, our category is equivalent to vector bundles
on the algebraic curve.  Mirror duals can be
found for Riemann surfaces of any genus.

If there is time, I will also discuss the action
of the mapping class group on the category.