Earthquakes

In work with Roger Barnard, I extended the discrete welding operation to Riemann surfaces. We gave a new discrete proof of Masahiko Toki's result that any conformal structure on a torus can transformed into any other by cutting and (unevenly) re-attaching. More importantly, we found a precise combinatorial recipe for deforming the geometry of one torus into any other.

I found a similar result for hyperbolic surfaces using combinatorial earthquakes. 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.

My student Eric Murphy has recently extended this work to complex earthquakes, as introduced by Curt McMullen. He has also developed an analogous theory for complex earthquakes on Euclidean surfaces.