Microlocal analysis on stratifications

Bernhard Gramsch

 

Algebras of pseudodifferential operators are discussed on singular spaces with functional analytic commutator methods, Lie algebras of vector fields and attached flows. This are localisations and quantisations on ramified manifolds (anyfolds). We get new results on local regularity and propagation of singularities, initiated by Sato, Hörmander and Egorov. Some motivations for the talk come from results of Ammann, Lauter and Nistor (e.g. Ann. of Math. 2007) and Schulze et al. (e.g. EMS Tracts 2008) and also from Waelbroeck (1954). Relations to the Oka principle are presented together with direct homogeneous Fréchet submanifolds in connection with classes of Fredholm operators. The related difficulties with the implicit function theorem for Fréchet spaces are avoided by rational methods.