I will discuss formal quantizations and their relation to the filtered ones and then the localization procedure for quantizations (sometimes called the microlocalization).

A hyperbolic structure on a surface is described by a representation of the fundamental group into PSL(2,R). Higher rank Teichmüller theory aims to go beyond hyperbolic geometry by studying moduli spaces of representations into bigger Lie groups, most quintessentially SL(n,R). I will discuss a SL(n,R) version of one piece of hyperbolic geometry—Thurston’s compactification of Teichmüller space. Boundary points of Thurston’s compactification are measured laminations: certain analytic objects generalizing simple closed curves. I will discuss a compactification of the SL(n,R) Hitchin component which is constructed in much the same way, and whose boundary points are geodesic currents which generalize closed curves with more intricate restrictions on self-intersection.