TITLE: What are some Applications of Teichmüller Theory?
QUESTION [19 upvotes]: I'm trying to collect some specific examples of applications of Teichmüller Theory. Here are some things I have collected thus far:

No-wandering-domain Theorem (Sullivan)
Theorems of Thurston (Classification of homeomorphism of surfaces, topological characterization of rational maps, hyperbolization theorems for special 3 manifolds)
Computer graphics. (Using the various metrics on the Teichmüller spaces as a substitute for Gromov-Hausdorff metric.)
String Theory (as elementary particles are modeled by loops, they generate a Riemann surface as they move through time.) 
Some applications to biology.(Brain morphometry)

Note: I am very sure that this is only a small fraction of what is out there, and I plan to continue to update this list.

REPLY [2 votes]: http://www.cacs.louisiana.edu/~mjin/ I saw a talk by Miao Jin where she used Teichmüller theory and Ricci flows to better store and analyse colonoscopy data. I believe her advisor at Stony Brook started a company which does that.