# Research

Discrete Conformal Geometry is a relatively new area of mathematics lying at the intersection of geometry and analysis. Central to this field are circle packings, configurations of circles with prescribed patterns of tangencies. The patterns are typically represented by abstract triangulations in which vertices correspond to circles and edges represent circles which are tangent. Packings thus have a combinatorial structure (the abstract triangulation) and a geometric structure (the size and position of the various circles). The interplay between packings like the ones below - packings the same combinatorial structure but different geometric structures gives rise to the deep analytic properties of circle packings.

For example, for a given finite simply connected triangulation, there are infinitely many different circle packings. However, the shared combinatorial structure of these packings allows the construction of maps between geometrically distinct packings. In the mid-1980's William Thurston conjectured that these maps are approximately conformal. Thurston's conjecture was soon proved by Burt Rodin and Dennis Sullivan, and the resulting explosion of research activity showed circle packings possess properties analogous to classical analytic functions and also produce approximations to classical functions in many situations where no other method is known.

It also appears experimentally, that circle packings for random triangulations also generate conformal maps in the limit. This is a fascinating phenomenon which is now only beginning to be understood.

In the 1990's, the study of circle packings was extended to Riemann surfaces when Ken Stephenson and Alan Beardon showed that every abstract triangulation of a surface can be realized by a unique packing (up to conformal automorphisms). In other words, a given triangulation picks out a unique Riemann surface that supports a packing for that pattern. Most Riemann surfaces are not packable, but a number of authors, including myself, have shown that packable surfaces are dense with respect to the Teichmüller metric.

One major component of my research then has been to understand these packable surfaces - how they are related to one another, how they are distributed in Teichmüller space, and how they can be manipulated.For non-compact surfaces, I provided a complete answer by showing all such surfaces are packable. For compact surfaces, however, the situation is much more complicated.

David Mumford and Eitan Sharon have recently developed a method of 2-D image recognition using conformal welding. Given a Jordan curve, they represent it using a conformal welding map. My work on welding circle packings provides provides the means to pass from the welding map to the curve and vice versa.

For questions, please email Brock Williams.