12/2/2010

Sa'ar Hersonsky
University of Georgia

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

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 2-cell 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 $(m-1)$ 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: http://www.math.uga.edu/~saarh/Papers/Papers1.htm (#18 & #19).