Riemann surfaces: a hyperbolic approach
Riemann surfaces are a central object at the confluence of many rich areas of mathematics, and one powerful legacy of Thurston’s work is the concrete study of Riemann surfaces via hyperbolic geometry. We take a hands-on tour through the fundamentals of hyperbolic surface theory such as trigonometric identities, geodesic length spectra, measured lamination theory as well as Thurston’s construction of optimal Lipschitz maps on closed and cusped hyperbolic surfaces.
Some very basic topology and differential geometry.
 Buser: Geometry and Spectra of Compact Riemann Surfaces;
 Thurston: Minimal Stretch Maps Between Hyperbolic Surfaces (ArXiv preprint).