Square tilings and networks

Richard Kenyon (Brown)

Abstract :

In 1903, Dehn showed that an aXb rectangle can be tiled with squares iff a/b is rational. What about other polygons or flat surfaces? This (open) question has connections with planar resistor networks and extensions of quadratic forms. I'll discuss the answer for tori and polygons with small numbers of sides, partially based on work on planar networks of Yves Colin de Verdière.