Boundary Value Problems on Planar Graphs and Flat Surfaces with integer cone singularities.
Boundary Value Problems on Planar Graphs and Flat Surfaces with integer cone singularities.

Sa'ar Hersonsky, University of Georgia
Fine Hall 314
Consider a planar, bounded, $m$connected region $\Omega$, and let $\partial\Omega$ be its boundary. Let $\mathcal{T}$ be a cellular decomposition of $\Omega\cup\partial\Omega$, where each 2cell is either a triangle or a quadrilateral. From these data and a conductance function we construct a canonical pair $(S,f)$ where $S$ is a genus $(m1)$ singular flat surface tiled by rectangles and $f$ is an energy preserving mapping from ${\mathcal T}^{(1)}$ onto $S$.The subject has an interesting history that started with Dehn (1903). References may be found here (#18 & #19).