My research has progressed in three principal directions:

1. Discrete Conformal Geometry
2. Applications of Circle Packing
3. Extremal Problems in Geometric Function Theory.

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.

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.  It is easy to show that there are only countably many compact packable surfaces.  By studying how changing the triangulation changes the resulting packable surface, however, I have uncovered the deep richness in the distribution of packable surfaces.

In particular, I have investigated the effects of conformal weldings, earthquakes, and complex earthquakes on packable surfaces.  Weldings are created by cutting a surface open and then gluing it back together unevenly.  I developed a combinatorial form of conformal welding, and in collaboration with Roger Barnard and my student J’Lee Bumpus have demonstrated how it can be used to manipulate the geometry of tori.  Earthquakes were introduced by Thurston and used by Stephen Kerckhoff in his celebrated proof of the Nielsen Realization Conjecture.  An earthquake on a hyperbolic surface is created by shearing isometrically along geodesics (the "fault lines").  I showed that earthquakes can be approximated by shearing circle packings; consequently, combinatorial earthquake deformations can be used to manipulate the geometry of hyperbolic surfaces.  With my student Eric Murphy, we extended these results to the complex earthquakes introduced by Curt McMullen.

A second major focus of my research has been applying discrete conformal geometry to problems which have defied classical numerical techniques.  My study of the combinatorial manipulation of packings on surfaces has had an important corollary – we have several methods for creating packings with prescribed geometry.  That is, we can manipulate the combinatorial structure of a packing to produce the geometry we desire.

For example, Thurston’s original conjecture used packings with the same combinatorics to create conformal (angle-preserving) maps.  We have described how to distort the combinatorial structure to create packings which induce quasiconformal maps (maps which distort angles by a bounded amount).  This has been of interest not only because quasiconformal maps lie at the heart of the study of Riemann surfaces, but also because the same technique allows us to create conformal maps of physical surfaces, such as brains.

Ken Stephenson, Phil Bowers, Monica Hurdal, et al have used circle packings to great effect to create quasiconformal maps of the human brain.  They use MRI scans of the subject to create an embedded triangulation of the brain.  This triangulation is then realized by a circle packing, producing a flat map of the brain.  Such a map is necessarily quasiconformal as the circle packing was created using only the combinatorial structure of the brain, not the geometry.  We have described a method to manipulate the triangulation so that the resulting packing will match the brain geometrically as well; that is, the map will be approximately conformal. 

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.

Finally, I have also been active in studying extremal problems in geometric function theory.  In work with Roger Barnard, Kent Pearce, Leah Cole, and students Casey Hume and David Martin we have

a.         Found the sharp bound on the Schwarz norm of hyperbolically convex functions.

b.         Described the extremal hyperbolically convex functions for Re  and Re .

c.         Described the extremal functions for the Fekete-Sezgo functional for hyperbolically convex functions. 

d.      Described the extremal functions for Fournier-Ma-Ruschweyh bound on  for bounded Euclidean convex functions.