Microlocalization and second microlocalization

Microlocal analysis is about localizing in phase space, i.e. physically in combined position and momentum, and mathematically in the cotangent bundle of an underlying space (the position space). For instance, one can ask where a function, or distribution, is singular (by various measures) and use this for a more detailed description of solutions of PDEs. This can also be a very useful tool in setting up a solvability/invertibility theory. Sometimes, however, one needs to microlocalize even more finely; this notion is called second microlocalization, which I hope to briefly describe towards the end of the talk.