Title: A rough but fast approximation to the Riemann map

When/where: Monday, February 9th, 2:30PM, 400 AKW

Speaker: Chris Bishop (SUNY-Stony Brook)

Abstract: Combining results from hyperbolic and computational geometry, we will show that for any simple n-gon P bounding a region O in the plane, there is simple, geometrically defined map of P to the unit circle so that
(1) the images of all n vertices can all be computed in time <= C n and

(2) these images are uniformly close to the true conformal prevertices (more precisely, there is a K-quasiconformal map of the unit disk sending ourpoints to the images of the vertices of P under a conformal map of O to the disk).

Moreover, C and K are independent of n and the geometry of P.

Thus we have a map which is easy to visualize, fast to compute (in theory), and gives a uniform approximation of the Riemann map. In addition, our map shrinks arclength along P, extends to a Lipschitz homeomorphism of O to the disk (with respect to the internal path metric), and has only localdependence on the shape of P (i.e., the cross ratio of the image of four adjacent vertices depends only on the boundary near those points).

The map is built using the medial axis of the polygon and the closeness to the Riemann map follows from a theorem of Dennis Sullivan's concerning hyperbolic 3-manifolds. We will describe the background, some applications and open questions, and sketch the proofs of (1) and (2).