Aron Wennman: Circle packings and Riemann mappings

Consider the hexagonal circle packing with discs of some small radius r. Use the boundary of a bounded simply connected domain D to punch out a portion, say P, of the hexagonal packing. This will serve as an approximation of D.

A theorem due to Koebe and Andre'ev states that one can rescale the circles of P without changing tangency relations, to produce a new packing Q in the unit disc, with boundary circles of Q being tangent to the unit circle.

In a sense, conformal mappings map infinitesimal circles to infinitesimal circles. Thurston suggested that if one maps the actual circles in the packing Q to the corresponding ones in P, and repeats this process as the radii in the hexagonal packing tend to zero, this will approximate the Riemann mapping from the unit disc to D.

This talk centers around Rodin and Sullivan's subsequent proof of this conjecture.