On discrete version of curve shortening flow One can define an analogous version of the curve shortening flow for polygons in $\mathbb R^2$, namely defined by the differential equation $\dot{p_i}(t)=\frac{v_i(t)}{|v_i(t)|^2}$, where $p_i$ is the $i$-th vertex of the polygon and and $v_i$ is the vector that goes from $p_i$ to the circumcenter of the triangle defined by $p_{i-1}$, $p_i$ and $p_{i+1}$.
The Gage-Hamilton-Grayson theorem states that simple curves remain simple under the curve-shortening flow.
Does this still hold for polygons under this analogous flow?
 A: Not exactly an answer to your question, but Peter Scott and I worked out a polygonal flow that is guaranteed to keep curves embedded in "Shortening Curves on Surfaces", Topology 33, (1994) 25-43.
A version of this is implemented in java at
disk flow applet
The problem with the Birkhoff flow is that long segments move faster than short segments, and can overtake them, creating self-intersections.  This can be overcome by getting the length of segments from the surface (using intersections with some sort of grid for example) rather than from a parametrization of the domain, as with Birkhoff.
A: This is not a direct answer, but may I point you to 
an earlier question of Ryan Budney,
and to the paper by
Bennett Chow and David Glickenstein,

"Semidiscrete Geometric Flows of Polygons."
  American Mathematical Monthly. April 2007.
  (MAA link)

 
Under their discrete flow, every simple polygon converges to a point whose shape is asymptotically an affine transformation of a regular polygon.
A: A related flow does keep simple, see Ileana Streinu's (Joseph's colleague at Smith!) work http://cs.smith.edu/~streinu/Research/robotics.html and related work by Connelly/Demaine/Rote
