TITLE: Applications of microlocal analysis?
QUESTION [28 upvotes]: What examples are there of striking applications of the ideas of microlocal analysis?
Ideally I'm looking for specific results in any of the relevant fields (PDE, algebraic/differential geometry/topology, representation theory etc.) which follow as natural consequences of microlocal analysis and are not known to be obtained easily without it.

REPLY [5 votes]: A pretty recent and impressive application of microlocal analysis is to analytic number theory or automorphic forms, via the orbit method as developed by Nelson and Venkatesh. In loc. cit. they set the stage for the theory specifically thought to be applied to problems in the area of automorphic forms (trace formulas, moments, subconvexity, and more widely problems on large families of automorphic forms).
A brilliant outcome of this setting is the proof of subconvexity for all L-functions of general linear groups, by Nelson.